I'll divide this guide into story-related achievements and missable/limited achievements.

I'll try to list the achievements in the order you get them. Puzzle images that contain spoilers are spoiler-tagged (those are NOT large blank spaces!).

Note: You need at least 2 playthroughs to get all achievements.

For your first playthrough, I recommend using hints (if you want/need), and skipping the spider puzzle, the number dials puzzle (the one with the numbers 12, 13 and 15), and the foe lionhead puzzle (you have to enter it, and actually click the button "skip button" when it becomes available, and do the friend lionhead puzzle only after you get the achievement).

For your second playthrough, you can focus on the rest of the achievements.

Farm Fresh
Find a way to enter your first painting


Sum of Its Parts
Successfully combined two items to make a third


Strawesome
Figure out how to help the farm


Let there be Light
Opened the Cloud Door using the heat of the sun


Romancing the Stone
Successfully opened the Heart Door


Where the Sidewalk Ends
Find a way to cross the broken bridge


Breath of Fire
Find a way to melt the ice on the other side of the room


Movin’ On Up
Find the hidden path out of the Fire and Ice room


Drawn to a Conclusion
Find Iris

Taking the Long Way
Successfully beat the game without skipping any puzzles


Limited! Conflicts with "Eye Hate Spiders", "Beam Me Up!", and "Foe Real?".
All you have to do is never click on "SKIP" during a puzzle throughout the game. Unlocked after beating the game.


Drawnslaught
Successfully completed Drawn without using any hints


Never click on Franlkin's portrait for a hint. Unlocked after beating the game.


Off to a good start
Beat the Foyer Chest Dial Puzzle without skipping


Don't skip the puzzle! It's this one:

Solution: Drag the pin on the right all the way down/right, and drag the larger disc all the all to the right to fit the slot. Then, drag the other pin down and realocate the smaller disc. Finally, turn the discs in order to complete the phoenix picture.


Stay Awhile
Successfully complete the Tile Puzzle in the Foyer Chest in 16 or fewer moves


The description is pretty straight-forward.
This is how the image show look in the end:
Solution:



Pane in the Glass
Repaired the broken window before the skip puzzle timer had a chance to charge


The timer only starts after you put the last piece on the window, so you might wanna experiment putting all pieces but one. Note that some pieces, while identical, will only fit in its exact spot, not in its "twin" piece's spot. If you still want the answer, check below:
Solution:



Over Achiever
Successfully found the magic staff in the hut before the shrunken head told you where it was

As soon as you enter the Hut Painting, you can even do it before making the witch appear, but you just can't get the hint from the shrunken head. Here's how: Pull the rope on the ceiling three times. It's where the cursor is indicating:



Toil and Trouble
Successfully completed the potion right the first time in the Hut Painting


The wording in the recipe might confuse you. That's why I'm here =)
Solution:



Eye Hate Spiders
Look away if it bothers you too much


Limited! Conflicts with "Arachnophobia", and "Taking the Long Way".
Skip the spider puzzle at the base of the Giant Tree.


Arachnophobia
Beat the Spider mini game in the base of the Giant Tree in 4 or fewer moves


Limited! Conflicts with "Eye Hate Spiders".
Take your time and think before you click. Here's what worked for me (click the Xs)
Solution:



Nailed it!
Successfully beat the Tree House floor puzzle before the skip puzzle timer fills up


This is a matter of getting used to the puzzle. The starting positions are different every time, so I can't really help you much. I'd say focus on a color at a time. Take a screenshot, pause to think about it, then do some progress. I'll provide an example below just in case:

Click the following nails, in this order: C-5-1-5-A-2-2-A-1-A-1-1-1


Finger Painting
Correctly paint the three clouds in the Tree Painting right the first time


Take your time in order to avoid mistakes. It's not hard if you know that you have to do it. Note that there are 3 clouds in the sky, so make sure you know what you're doing before you start drawing.

Just Filling Time
Successfully complete the Flood Fill mini game in the market in 13 moves


It's the one that gives you 4 colors in order to fill the spaces, not painting adjacent sections with the same color. It's easy if you know the solution, or figure it out before trying to paint anything. You can either use a logical approach and only fill a space if it's already touching other 3 colors, or take a screenshot and experiment if you want. Or you can look below.
Solution:



Happily Ever After?
Successfully created the Hero for the Clock Tower play


Just don't skip any parts of the hero puzzle! If you need help with the equipment part, check the next achievement.


Once Upon a Time
Arm the Knight right the first time


Solution: I - Shovel ; II - Metal Shield ; III - Heart
Do NOT skip any parts of this puzzle, not even the coloring part!


Beam Me Up!
You can beat it next time


Limited! Conflicts with "Number Crunching", and "Taking the Long Way".
Skip the Number Rings puzzle.


Number Crunching
Beat the Number Rings puzzle before the skip puzzle timer fills up


Limited! Conflicts with "Beam Me Up!".
Check a working answer below (the screenshot was taken right after the puzzle was solved, but the numbers are right)
Solution:



The Writing on the Wall
Translate the cave wall images


Don't skip the puzzle! It's not hard, just time-consuming. Check the image below for the answer.
Solution:



Speak, friend, and Enter!
Successfully open the gate to the Griffon’s Castle without making the gate’s eyes red at all

Limited! Conflicts with "Foe Real?".
When interacting with the painting with 2 lionhead beside it, pick the "friend" one, solve the puzzle normally. After the teeth get green, go to the gate inside the painting and arrange the door correctly, as to form a CIRCLE pattern. Don't interact with the Foe lionhead! Don't make any other patterns at the gate!


Foe Real?
Ignore the Lion that isn’t a friend


Limited! Conflicts with "Speak, friend, and Enter!", and "Taking the Long Way".
Description is deceiving! You have to interact with the foe lionhead AND skip the puzzle. You'll then need to push its nose to reset it, in order to advance with the story.


It’s drafty in here!
Successfully beat the Griffon’s Castle Window Puzzle in 5 or fewer moves


There are 3 eye "switches" on this puzzle: left (L), down (D) and right (R). You have to make the lit glass piece be where the arrow points to. I'm not sure if the starting position is always the same. If they're not, here's what each eye does:
L - 7 spaces clockwise
D - 3 spaces counter-clockwise
R - 5 spaces counter-clockwise
In my case, the light started 4 spaces clockwise from where it needed to be (ie, it needed to move 4 spaces counter-clockwise). The solution could be: LDDR (the order doesn't matter).


Eye for Detail
Successfully complete the Griffon’s puzzle without needing to restart


You have to complete the "Simon Says" puzzle without making any mistakes. If you're having trouble, you might think of the lights as 1, 2, 3 and 4 from left to right and write down which lights will turn up so you can input the right answers.

That's it! If you find any mistakes, please inform me!

If you think this guide helped you, don't forget to give it a thumbs up! =)