I do not like this game. I think the main mechanic is wholly inelegant and the game is tedious and overly punishing. But I'm a completionist. I finish what I start. I have put off playing more of Neon Warp for years, hoping someone would pick up the thankless task of writing a complete guide so I could look for assistance whenever I'd get stuck, but it never happened. And thus it falls onto me to write a guide for the probably single-digit number of people who will ever read this, who either have tastes completely different from mine (good on you!) or who - like me - were foolish enough to buy this game (I pitty you).

Just to be clear on terminology: There is a board made of tiles. Every tile has one of three colours - magenta, cyan, or purple.
The goal of the game is to turn the whole board into the same colour except for up to 1 tile that is allowed a different colour. You cannot choose your target colour! Look to the level select screen or to the colour of the controls at the bottom of your screen to figure out your target. The "up to 1 tile" left at the end can be of any colour and it does not have to be adjacent to your cursor.
You pick your starting location (click on a tile or use arrow keys and select with Enter). Picking a starting location does not count as a move, but it triggers whatever special effect the picked tile might have.
Once you have picked a starting tile, every move of your cursor to an adjacent tile triggers the following colour changes: If origin tile and arrival tile have the same colour, nothing changes. If origin tile and arrival tile have different colours, the arrival tile will change to the third colour.
Tiles with markings have special properties: Horizontal and vertial bars affect the two tiles horizontally or vertically adjacent, and the plus tile does both, affecting all 4 adjacent tiles. The last mechanic is denoted by an empty square, which attaches a colour to your cursor, overriding normal colour changing rules for the next 3 moves. All special properties use the colour of the trigger tile *after* your arrival. So if you move from cyan to a magenta plus, it would turn the plus tile itself and all 4 adjacent tiles purple.

The starting location matters a lot. Some starting locations make the level unbeatable. If you are stuck, maybe re-evaluate your starting location.
Most levels have very few "sensible" starting locations. There's no use starting in the middle of a group of same-coloured tiles since your ability to change colours depends on moving between adjacent tiles of *different* colours. Also be aware that many levels are rotationally symmetrical. So it may look like there are a lot of different starting positions, but many are the same, just rotated 90° or 180°.
Be aware that if your starting tile has a special mechanic you can trigger it for free (on move 0). This is often a good way to get a head-start.
Long distances are your enemy. You are judged by a move count, so make as few "empty" moves as possible. If you can, start at one edge of the board and work to the other. Finish areas before moving on, so you don't have to return. Distances are especially punishing if you have to carry a tile with you (moving a coloured tile along takes 3 moves for every 1 tile advanced), so try to avoid that (it's still a pretty common mechanic though).
Special tiles are very powerful since they change more than 1 tile at a time and the colour of those other tiles does not matter at all. Typically (as in >90% of the time) you want to trigger a special tile with your target colour. Once you figure out how to set that up, it's usually a good way to beat the level (but beware - some levels force you to make non-sensical or counter-intutive moves).
Special tiles can also be used to "take out the trash". If you have an isolate tile left over, it's sometimes faster to park it beside a special tile and annihilate it rather than carry it with you to another coloured tile.
If you are getting stuck on larger levels, try to isolate parts and optimize them individually. Writing this guide actually helped me through some of the more difficult levels because I had notes and could compare approaches, thus figuring out how I may be able to shave off some moves.

Neon Warp is incredibly unforgiving. Levels have 3 tiers of move goals (denoted by 1, 2 or 3 yellow squares), and if you can't meet the most lenient one (1 square) you don't get to advance to the next level, even if you can eventually complete the level. If you get stuck on a level, the game wants you to beat your head against a wall. Now you can either use the guide below to have the solution handed to you. Or you can bypass the level and return later with a fresh mind. Unfortunately for this we have to edit the save file.
On Windows the save file resides in "C:\Users\[your name]\AppData\Local\Neon_Warp\save.ini". Open the file in any text editor (even Notepad is fine) and the structure should be immediately apparent. It's just an unordered list of two-line entries consisting of a level number in square brackets on the first line and a squares value on the second line. To mark a level as beat, just add its number in square brackets to the bottom of the file on a new line and then put on a second new line: squares="1.000000"
That way you can keep playing and you also know to return to the level to eventually clear it, since it has only 1 of 3 squares awarded. Don't feel too bad - as best as I can tell the move goals are wholly arbitrary. Sometimes 1-square has way too little slack to even allow for fairly elegant simple solutions. And sometimes even 3-squares can be beaten by more than 10 moves. Not being able to immediately meet the goal for 1-square does not automatically mean you're bad, more often than not it means the game is badly balanced.

If skipping levels is not enough and you don't care about doing it all yourself - worry not, what follows is the full solution to all 150 levels. These are not always optimal solutions, but they all meet the tightest move goal or even beat it. Just comparing with hdgnomus' guide for the first 17 levels some levels there already have better solutions than mine. If you find better solutions yourself, you can add them in the comments if you want.
The notation for moves uses only first letters: Right, Left, Up, Down. Sequences are arbitrairily chunked using spaces to make them easier to read. There are no pictures and I know the formatting is poor - I really did not want to spend more time on an unenjoyable game like this, I'm sorry.

1: start in the top left corner, RLD RLU
2: start in the top left corner, RRD LUD UDR
3: start on the second tile, LRR LRLRR
4: start one tile below the very top, UDUD RDDU DULR UDUL
5: start at the top left corner, DRDD LRUDU UDUL
6: start at the far right, L [tutorial for bars]
7: start at the right-most cyan tile, DUL
8: start in the bottom left corner, RRUU
9: start at the top magenta tile, LUDR LRD UDR DRLRL ULRD
10: start at the top left corner, DDRRR UULRL LRLL DUR

Achievements:
Triple star - Complete a level with 3 stars.
Good start - Complete 5 levels.
10 outta 10 - Complete 10 levels with 3 stars.

11: start at the bottom left purple tile, LLUR LRLRD
12: start at the far left, RLR DRL UUD RDUL
13: start at the bottom magenta tile, DUD RLRLR RLRUDU UDULRU DULRLL
14: start on the first cyan tile, LRLL RRLRLR RRRR
15: start in the top left, DRLR URDUD RLL (3 under par)
16: start on the right-most cyan tile, RLLLRL RRRLL RLRRR LLRR LRR
17: start in the very bottom, ULRLR RLUDUU
18: start at the top, DR RLR DU RLR D
19: start in the top left corner, DRRRR [tutorial for pluses]
20: start at center, DDUU RRLRLL DU RLU

21: start on the left bar, RRUL DUDURR (1 under par)
22: start in the top left corner, RLD URR LRLRD
23: start in the bottom left corner, RLR UURRRR LRL DUD DUDUD LRLL RLLUDR (1 under par)
24: start in the bottom right corner, UUULL DUDU DDDU [tutorial for squares]
25: start on a special tile and just make 2x2 squares so you always end up beside a new special tile after 3 moves
26: start on the left bar, DURL UDRLR RDUDUU
27: start on the top left plus, URR DDD LRLRL LLUUUD RLDD RLR URU (5 under par)
28: start in the top left, DDRR UDUDD
29: start on the left edge one tile from the bottom, URRDDLL UUDU RLDU DDUD RLRLR RLRU
30: start on the left-most square-tile and just follow the path only going forward, at the right-most square-tile turn around and go back two tiles

Achievement: Four-wheel drive - Complete all 4x4 levels.

31: start in the corner of the middle square, DUDU RRDDUD LRLRL LRL UDUU
32: start on the far left, RRUDUD DULRR
33: start on a cyan tile, move to the middle, then trigger both vertical bars
34: start on the top plus, DDUU LRLR UDUD RLL
35: start in the middle, DLDUDU RRDU LLRRL UDUDUU LUDUD RRUD LLRRLD (1 under par)
36: start in the top left, DDDRR UUULLD URRRRDD
37: start in the top left corner, RRRLLL RRLR DUDDUD DDLRUU LRDDRL UUDUR
38: start at the far left, RLRR LRRRLL UUDDDDU RLUUD RDUL (1 under par)
39: start at the top left purple tile, DUDRLD UDURLR RRR UDUDDU LRLL RLL
40: start at the bottom left corner, URRRRRD RRRR URRRR (2 under par)
41: start at the top center, DUDUDD
42: start at the top left corner, RRRLL RLLDD UURLDD RRRRDU DDUUDD LLLRR LRRLRLRU (4 under par)
43: start on the top left plus, LURD LUDUD RRD (3 under par)
44: start on the left plus, ULRLRR RRDULD DRLRLL (1 under par)
45: start in the top left corner, RDUDD DDUDU RRDU DUUL
46: start in the top left corner, DUD DUD DUD UDDUU DDRRRR LLU
47: start on the left bar, URLDD UUDU RLRLR RLRR LRRU
48: start in the bottom left corner, UUDUU DUURR LLD URRRR DDUDD UDDLLR
49: start on the magenta square, LDD UUUU RRR
50: start on the top plus, LDUR DDD

51: start on the left-most cyan tile, DUDD RLRR UUDD RLR RLRLR ULLL (1 under par)
52: start on the bottom-most purple tile, RLRLL RUU LRLRR
53: start on the top-most purple tile, LUUDUD LRDRR UUDUD RLL (5 under par)
54: start in the center on the horziontal bar, ULRLD
55: start in the top left corner, DRLRR DUDU RRLL DDLRRR
56: start on the left bar, DLRLR UDU UDU RUUDDL
57: start on the top-left cyan tile, UDUR RDRD UL UDU LRL LRL DUD LRLD DRDU, move around clock-wise to the other cyan tile, DL (1 under par)
58: start on the left bar, URLRR DLD RLRLL
59: start on the bar in the center, DDUUU DDUD DUDUD RLR RLR UDU UUUDDD
60: start on the bar in the center, URLDD UDUDL
61: start on the second tile of the bottom row, LRLR RLRU URLRLL
62: start on the middle tile in the left-most colum, RDU LRUD RLRLRR
63: start on the top bar, ULRDUD RLRD DLDLR UDURLR UDURLRR
64: start on the top cyan tile, RDL DUDL
65: start on the top cyan tile in the left-most column, UDDD UUURR DDUDD UUURR DDUDDU (easy to accidentally move 1 tile too far in the end)
66: start on the fourth tile in the bottom row, ULL ULDD RRLR RUD RLL RLR RRLL RLR RUDL RRLRU RDLRR UUDUL (1 under par)
67: start in the top-left purple tile, LRLRU RUDUD RD RLRLD LRLD URLU
68: start on the bottom cyan tile of the right column, DUDL RUUUU LRLR DUD DLL (1 under par)
69: start one tile to the right of the left-most cyan tile, LRL RRL RRL RRL RDUD RLRL URRR (1 under par)
70: start one tile to the right of the left-most cyan tile, LRUDD RLR RLU DDR URR UUDR (7 under par)

71: start on the left square, LDUR UUDUD RRDRD
72: start on the left plus, UUDD RLRR LLL RUDUU
73: start in the very bottom, URUU LDL (3 under par)
74: start in the top left, DDDD URRRR DUUUU DLLL RDUL RLR DUD DUDR
75: start on the top center bar, RDDLL DLRLR UDU RLR RLR DUDRL UDU UDU UDU URL DDUD LRUU (3 under par)
76: start on the second tile of the top row, DRLUD RLRRL DUDD ULD (4 under par)
77: start in the top left corner, RDD RRUU DUL
78: start in the bottom center, ULUU RRDL
79: start on the middle purple tile, RLUDUD DURRLL RRUURRD
80: start in the bottom left corner, RLUU RUURR DDRDD LLLL UURUUR (2 under par)
81: start on the middle tile of the top row, DLR UDRLD RRDDLL RLLRR (3 under par)
82: start on the top bar, ULRDD LLLRRR UUDULR DUD DUDD
83: start on the second cyan tile from the left, DRLLL UURR LRLL DDRR LRUDU UDUDU RRR LRRR RRR DDLL RLRR UULL RLDUD DUDUD LLL LRL LRU (6 under par) [one of the hardest levels in the game in my opinion]
84: start on the left-most tile of the top row, DDLDD UDU UUDDU RLDR (left side done) UUURRD RLRLDD RDULL DUDUU DUU RRDDU LRLL RDUDL (8 under par)
85: start on the left plus, DDUDUU LUDR RLRU DLDD UDU UDU LRLU (2 under par)
86: start in the middle, ULLRR LRLR DUDD UUDUR (3 under par)
87: start first row second tile from the top, URLD URRRR LRDU DDDD UDLR LLLL RLUD (2 under par)
88: start in the middle, LRDU RLU DRU
89: start on the bottom left square, DUUR URUDDD RLRL LRL LRL UDU UDUL
90: start at the left most cyan square, LDD UURR UULL RRDD LLRL UDRR RLDD UURR DD LRUU RLUU DDRR UUL RDD RLDD UURR DDLR UUUU (4 under par)

Achievement: Simple as 5x5 - Complete all 5x5 levels.

91: start at the top-left corner of the right-most magenta square, UUU LRLR UDLL DL
92: start at the fourth square in the top row, DLDDRU
93: start at the left square, ULLL DUDU RDDD LRLRR RRR UUL URRR DUDU LDD (3 under par)
94: start on the second tile of the second row, LDRU LRR LRL DRRR LRR LLDUDUU (2 under par)
95: start on the top left cyan plus, DDD UUR ULRLR RLRR LD
96: start above the first purple colum, DDDD UUUU DU RLR RLR DDD UUU RLR RLR RLR RLR DDD DUDUD DUD LRL LRL UUU
97: start on the second tile of the top row, RRRR RRDRR LLULLL RLRLD DDRU RRUL DRR DLRLRL LRRD RRUL RRUL RRUL RRUL (2 under par)
98: start on the cyan square, DDDD UUU UDUD RRLRR LLLD ULRLLL (2 under par)
99: start on the third tile of the bottom row, URLD UULR DUD UULR DUDUU DRR LUD RRLD (3 under par)
100: start on the second cyan tile from the left (not the square), URLD URUR UDU RLRR RLRD RLR DUD RDRLRDU (2 under par)

Achievement: Benjamin Franklin - Complete 100 levels with 3 stars.

101: start in the bottom left corner, URR URUUL URD DUD DUDL (1 under par)
102: start on the top purple square, DUDR DULU
103: start on the bottom one of the left-most magenta bars, ULDU RLR UDRUR ULR DUDDU RDD LDL (3 under par)
104: start on the second square of the top row, DRULL DDDD LRRR DULL RRLRD RUUUUR
105: start on the left-most edge on the second tile from the top, RRDD LRLR RRRR RRUDUD LULUU RD (3 under par)
106: start on the sixth tile on the bottom row (below a cyan bar), ULLRR ULDR RUDL RUDU UDUU DLDD RRR RRLLL RUDU RLULU DR (easy to accidentally move 1 tile too far in the end)
107: start on the right-most cyan tile, LRL DUDUD LRLRL LRLUD DL LRRUD LRL LRL DUD LLRR LRL LRL DUD LLRR LRL LDL RURLRL UDD (11 under par)
108: start on the right-most cyan tile on the bottom row, LLLRRRR UUU RRR UULL RLLLRR
109: start on the bottom row below the plus, UUUDUR LDDD UUDU LRLL
110: start on the first tile of the bottom row, ULL UUU RRR RRR RDDD LLL UL

111: start on the single purple tile in the bottom left, RUL RUU DDRUU DDR LLRD LD
112: start on the second tile of the second row from the bottom, DRRUURU RLRR LLRLR RLR UDU RLDR RRDUD DURDDR
113: start on the bottom left bar, LLDDD UDU UDU UDU RLR RLR DUDUD RUUL (10 under par)
114: start one tile below the cyan square, ULU RDDR
115: start on the magenta tile in the first column (third from the top), UDRR LLUU RRDU RLR RLR RLR DUD DUD DLL RRU DLLDD [gave me some real difficulty]
116: start on the second row on the second tile from the left, RRR DDD LLL UU RUR DRD LDUU DLRLR LUDR (4 under par)
117: start in the top left corner, DRRR LLDDD LUU DDRR DLL URRR DRR LURU LLL
118: start above the bottom-left cyan plus, DURD LLRL RDUU UUR
119: start to the left of the bottom cyan bars, RLUDRR LLUUR
120: start on the second tile of the top row, DUL RDUDUD RLRD RRDDUL RDUDUD RLRD (1 under par)
121: start on the third tile from the top in the first colum, RDLU RLU RRUL RDUR DLRLRD LLRLRD RRUR DLRD LUDUDD LLUR (4 under par)
122: start on the second tile in the first row, LDDDD URR UURDD RURU DDUD DUUUU
123: start on the second tile of the top row directly above the cyan plus, DUL DDR URULL RRLDUD RLD DUD DUD DUD RLR RLR RLR RLR RLR UDU UDU UDU RUDLU (7 under par)
124: start on the third tile of the second row directly above the magenta plus, DRUDD LDRR
125: start on the magenta tile left of the cyan square (second colum), RLDDL RLR UDU UDU LRR LRR LURRUU RLRL DUD DUDR
126: start on the third tile of the bottom row directly below the magenta vertical bar, ULRD UDU UUU DRR DLRL RLU DLRL UDU UDUL
127: start on the left cyan square, DDRD RLLL RUUU DURL (easy to accidentally move 1 tile too far in the end)
128: start on the first colum at the second square from the top, DDDD RRU DUD RRL URDUD RUUL RLR UUD LURLR ULLD UDU LLR DLUD UDD
129: start on the third tile from the top on the left-most colum, UDRR DRRR UDD UUUD LRD LRL LRL LRUDL LRL LUDR LD (5 under par)
130: start on the third tile of the bottom row, RLU DR LRL URUL ULULUL

131: start on the third tile of the bottom row, UDLL UUDD UD RLR RLR UURLDD RRLU DRR UU (4 under par)
132: start on the magenta horizontal bar in the second row from the top, LLRRR DDD UUDD UUDDD UUD LRL LLR RRR
133: start on the fourth tile from the top on the second colum (cyan corner), LDURL URRR LDUR LRR DRD ULRU (1 under par)
134: start on the second tile from the top on the second column (beside the magenta plus), DLRU DDUD URLRU RRRR UDDU LLDR
135: start on the first cyan tile on the top row (above the magenta bars), DDU LLL ULD LLLLL RUDD UDU LRDL
136: start on the third tile of the top row, LLRDL DDDD URD RRRR UUDLU URULD LD (1 under par)
137: start on the bottom row cyan square, ULU RUDR RRU DLD RDU RRR DURU LUD
138: start on the left cyan bar on the second row from the top, RDLDR RRLU RUUL RLR LLR LLR DDD LLL RUL UUR LRL RR (2 under par)
139: start on the first tile of the second row from the top, UDUDD UUR DUDD DDRU DLLL DUDUU (3 under par)
140: start in the top left corner, DDR DLD RDR URDR URUL URUL ULDL
141: start on the third tile of the second row from the bottom, RULD LLDR LRL RUD RLRLR RLRR RRR LLUUD LRRR
142: start on the fourth tile of the second row from the bottom, LLDR LRLL UURLRR RRUU RDLRLL UURLRR
143: start on the first tile of the second row from the top, UDR DUDU UDD RRLL DRR LLDU DUDD UUU LRL LLRR DLLRR DUDUDD (3 under par)
144: start in the top left corner of the cyan square, LLRD RRUUU DLDD RRRL ULLDD DUR (5 under par)
145: start on the fourth tile on the bottom row, RRLLU DRL RLL RRUU RLLU DRL RLL RRUU RLLU DRL RLL
146: start on the fifth tile on the top row (second magenta bar), LLLL RRR RRR RRR LLL LLL LLL DDDDD RLR RLL RLR RLR RRR LLL RLR RLR RLR RLR RRR UDUUU (5 under par) [one of the hardest levels in the game due to the counter-intuitive start]
147: start on the first tile in the top row, DDUD LRLR DLD URRRR DUDU RUU (2 under par)
148: start on the lone magenta square in the second colum (third from the top), UDD DUD LRUD RRR DU LRL UDLU RUU RUD UDR (8 under par)
149: start on the second-to-last tile of the top row (last cyan bar), LDDDD LLLL UUUU LLLL DLLLU RDDDD RRUURR DUDUU (3 under par)
150: start on the fourth tile of the top row (first purple tile), LRDL LUL RDDR LLU RDDR RUR RUR ULLD RRD LDL DRR DLLU LDU LLU DLU RDDR LLU (6 under par)

Achievements:
The Game - Complete all levels in play mode.
Yin and Yang - Complete all levels in play mode with 2 stars.
Neon Warp - Complete all levels in play mode with 3 stars.

Time trials are randomized boards using all basic and special mechanics from the pre-made levels. It's probably best to only try them once you have at least learned about the square mechanic on level 24.
You get a set time to start out and it counts down the seconds from there. After solving a full board you get a fixed amount of time added to your clock (how much depends on the size of the board) and a single point to your streak counter (those are the points needed for the achievements).
I'm not a huge fan of speedrun achievements to begin with, but time trials for a puzzle game is a new low. Thankfully fusecavator in their guide has discovered that the game is largely CPU-capped. Their approach has been to add a bunch of debug options, slowing down the game. But this is an inelegant way that also degrades your hardware by writing gigabytes of useless logs. Thankfully there is a smarter way. Equipped with the valuable knowledge that the CPU speed matters we can turn to a little tool called Battle Encoder Shirasé[mion.yosei.fi] which provides an easy way to CPU-limit just a single process. Use at your own peril - it worked brilliantly for me, but it might not do the same for you. Simply choose "Neon Warp.exe" from the list of processes to target it, and then scale *up* the limiter until you notice an effect (most notably the sound will become glitchy). At -70% the game ran slowly enough that I would gain roughly the same amount per stage that I had spent solving it, but since it depends on your hardware, your sweet-spot number might be different.
Be aware that you will only lock in the new highscore (and thus get the achievement) if you eventually fail the time trial! You cannot rack up beaten stages and then quit to menu; you will lose all your progress! So if you use the CPU-limiting method you might want to eventually turn off the limiter once you have enough points so the time-out happens more quickly.
"But isn't this cheating and defeating the purpose of time trials?" you might ask. Why yes, yes it is. But those time trials are silly and some of the achievements for them are insanely difficult if not impossible. Take note that even the developer of the game has positively commented on fusecavator's guide linked above. I have a feeling these achievements are not really meant to be gotten without assistance.

Achievements:
Harder - Score 10 points on 4x4 time trial level.
Better - Score 7 points on 5x5 time trial level.
Faster - Score 5 points on 6x6 time trial level.
Stronger - Score 1 point on Madness time trial level.
It's a Mad, Mad, Mad, Mad World - Score 5 points on Madness time trial level.