The guide is complete guys! Any doubts and/or suggestions, please leave a comment :)
Besides trying to add the most info i can to help you on all achievements, i highly recommend you to follow a good walkthrough on this game, first to not miss any of the many optional events in the game (some will lead you to some achievements, so they are important) and second to enjoy all the content this classic RPG has to offer. And please, DON'T forget... Turn up the volume while playing this :)

The Awakening
Wake up after having cleared the labyrinth for the first time in the dream.

Cannot be missed.


The Nemesis
Defeat Lynx at Fort Dragonia.

Cannot be missed.


The Dream Labyrinth
Re-visit the labyrinth in Fort Dragonia—this time, for real.

Cannot be missed.


The End and the Beginning
Arrive in Chronopolis, the Time Fortress.

Cannot be missed.


The Enemy of the World
Defeat Dark Serge at Fort Dragonia.

Cannot be missed.


The Anomaly
Use the Dragon God relics at the Dead Sea Ruins to enter the Sea of Eden.

Cannot be missed.


The Defiant
Defeat FATE, the mother computer of Chronopolis.

Cannot be missed.


The Devourer Undone
Defeat the evolved form of the Time Devourer.

Cannot be missed.

Flight of the Swallow
Use Serge's unique level 7 tech, Flying Arrow.

You will get it after you get your 35th boss star. Serge won't be available when you get the 35th star (more details on the spoiler tag ahead), but as soon as he gets available, get into a fight and use his new 7 tech Flying Arrow (if it's not automatically placed in his element grid, you need to equip it on Serge's element grid) You will get your 35th star while hunting the dragons for their relics. On this part of the game you will be playing as Lynx, so as soon as you get Serge's body back, do as explained above


Call of the Void
Use Lynx's unique level 7 tech, Forever Zero.

You will get it after you get your 35th boss star, while doing the Dragon's Relic Hunt quest (if it's not automatically placed in your element grid, you need to equip it on Lynx's element grid after getting the 35th star)


Lunacy
Use Harle's unique level 7 tech, Lunairetic.

You will get it after you get your 25th boss star. (if it's not automatically placed in your element grid, you need to equip it on Harle's element grid after getting the 25th star)


Lucca's Legacy
*by missable i mean that it's missable on that game if you dont get it on that single oportunity, but if you miss it you still can get it on a New Game +*
Use Kid's unique level 7 tech, Hot Shot.

In the first room of the burning orphanage, check the machine and push X, L1, triangle, R1, triangle and you'll receive it.
Edit: Thank you so much Kitty for you video and comment, i will post it here so people can check how to get it on this Remaster: https://youtu.be/j5aEDfGKzO8


Deadly Delta
*by missable i mean that it's missable on that game if you dont make all the specific choices to get an "optional" character and her tech as well, but if you miss it you still can get it on a New Game +*
Use the triple tech Delta Force.

To do this Triple Tech, you need Serge's level 5 tech - Luminaire (cannot be missed), Leena level 7 tech - MaidenFaith (MISSABLE) and Razzly level 7 tech - RazFlower (MISSABLE)

Leena's level 7 tech : getting Leena's level 7 tech is simple: at the beggining of the game, when you first reach Opassa Beach after collecting the Komodo Scales for Leena, you will have two options to choose while talking to her. On the first one, you must tell Leena that you remembered her promise and on the second one that you would never forget this day. Then, later in the game after obtaining the Tear of Love, talk to Leena's grandmother in Arni Village (home world).

Razzly's level 7 tech : besides being missable, this one is very annoying to get, so please pay attention to all the steps (a lot spoilers ahead):
First of all, when Korcha asks Serge if he will save Kid after she was poisoned, choose to save her ("find a way to save her" option). If you choose to not save Kid, you can't recruit Razzly! After choosing to save Kid, you will go after the Hydra Humour item on Hydra Marshes (Home World). Once there, free Razzly from her cage, but DON'T include her on the boss fight with the Hydra at that same place. Again, if you defeat Hydra WITH Razzly in the party, you CAN'T get her level 7 tech on this game anymore!
By not taking her to Hydra fight, her sister will be killed on the invasion of Water Dragon Isle (Home World) - this event can be done after the events with the Ghost ship/S.S. Invincible. You can go to Arni Village (Home World) and pay 100G to the old fisherman at the docks of the village so he can take you to the Water Dragon Isle (since you won't have a boat on Home World at that moment). On that island, take enogh time to kill all the dwarves. By wasting some time while on the island, you make sure that Razzly's sister, Rosetta, get killed (you need it to happen).
After the Chronopolis events and also after Terra Tower arise, return to Water Dragon Isle (Home World) and speak to the fairy in front of the regrown tree at the center to get the tech.


Z One and Only
Use the triple tech Z-Slash.

To do this Triple Tech, you need Kid's level 5 tech - RedPin (cannot be missed), Serge's level 7 tech - Flying Arrow (cannot be missed; for any specifc doubt, check Flight of the Swallow achievement info above) and will need Sprigg to Doppelgang the Slash enemy, a boss found at the Bend of Time (island southeast of the main island, following the way to Marbule), but only on NG+. Remember that you either need to kill Slash boss with a hit from Sprigg or of some other character using Forget-me-not-pot accessory.

An Unchanging Future
See the bad ending.

Finish the game after defeating the Time Devourer without using the Chrono Cross properly, just by getting his big HP to 0.


A New Future
See the good ending.

Finish the game by defeating the Time Devourer using the Chrono Cross after using elements in the following sequence: yellow, red, green, blue, black, white during the final battle.
It requires a little practice to do, but it worth the trouble o/

Once you finish the game for the first time and start a New Game + (NG+), you can fight the final boss at any time, just go to Opassa Beach and use the Time Egg at the purple portal. You can do all the NG+ endings in a single NG+, you just need to save before the correct moment to do that specific ending, do it, get the achievement, load the save and keep playing until you reach the next one.


The Developers' Room
See Ending 1: The Developers' Room.

To get it, you need to kill the final boss before talking to Leena on the Pier in Home's Arni Village or recruiting Poshul. Just start the NG+ and run to Opassa Beach. This one is boring to do on the first NG+, since you need to defeat it with Serge alone. I would suggest to do it on the second NG+. You should take a look at The Ultimate Weapon achievement as well, will really help to do this ending.


The Acacian Empire
See Ending 2: The Acacian Empire.

For this one, you need to finish the game after talking to Leena on the Pier in Arni Village (Home World) or recruiting Poshul, and before breaking into Viper Manor.


The Magical Dreamers
See Ending 3: The Magical Dreamers.

You need to finish the game after recruiting Razzly in Hydra Marshes (Home World), choose to save Kid and get the Hydra Humour, and before completing Fort Dragonia (Another World) and also before Lynx becoming a playable character. Nikki and Razzly must be in the party.


The Shop Staff and the Three Unlikely Heroes
See Ending 4: The Shop Staff and the Three Unlikely Heroes.

Finish the game when Kid is still poisoned.


The Rebirth of Marbule[/previewicon]
See Ending 5: The Rebirth of Marbule.

Finish the game after Lynx steals your body and before saving Riddel.


The Last Stand
See Ending 6: The Last Stand

Finish the game after saving Riddel in Viper Manor (Another World) and before sleeping at Hermit's Hideaway (Another World) and then triggering the events leading to Harle leaving the party. PAY ATTENTION to this one, as the time frame is really small and you can miss it if you dont pay attention and advance into the game!


The Viper Orphanage
See Ending 7: The Viper Orphanage.

Finish the game after defeating Dario and getting the Mastermune and before defeating FATE at Chronopolis.


The Dark Fate
See Ending 8: The Dark Fate.

Finish the game after Harle leaves the party on the S.S. Zebless, and before defeating FATE at Chronopolis.


The Dragon God's Revenge
See Ending 9: The Dragon God's Revenge.

Finish the game after defeating FATE at Chronopolis and before defeating the Dragons God temporary form in Terra Tower.

Immoral Compass
Get 10,000 points in the compass roulette mini-game.

I suggest to take a look at Wash's guide about the minigame, as his suggestions are the same as mine xD

Also, as suggested Hamtaro Hamaguchi on the comments:
while you can't enable slow/fast speed boosts while in the room, you can press the ESC key while in-game to bring up the launcher's settings which basically pauses the game. I figured this out when I got to the part of the game with the casino and was expecting to basically pause buffer the input to cheese the mini-game until I found that you couldn't pause or use speed boosts while in the room.

Grand Champion
Complete the Grand Slam.

After the events on S.S. Zelbess to unmask Fargo, the Grand Slam can be accessed inside the ship, where you see a cage and some mosnters fighting. The monsters you use in this event are the same monsters you learn to Doppelgang with the character Sprigg. Basically to learn to transform into a monster with Sprigg you need to kill that monster with her (she must do the killing blow). Since Sprigg isn't so strong physically, another way to learn to transform into monsters is to get an accessory called Forget-me-not-pot. You can get 2 of these: one in Chronopolis, right of the room before getting into the main elevator; and the second you can steal from the boss Peppor on a sidequest you will need to do to reach the optional boss Dario, who will give you the achievement The Second Coming. With this accessory, you can equip it on any character and if that character kills a monster that Sprigg can transform into, it will go automatically to her Doppelgang list of monsters, even if she isn't at the party.
An easy way to do this event is to wait until you first NG+ and then kill all the 3 bosses (Ozzie, Flea and Slash) in the Bend of Time (small island between the main island of the game and Marbule) with Sprigg or with a character with the Forget-me-not-pot equipped as mentioned above. This will not only make Grand Slam very easy since these 3 bosses are really strong, and either way you need to learn to Doppelgang the boss Slash to do the Z One and Only achivement, so this is probably the best oportunity to do it. Since Grand Slam has 3 rounds of fight, put a strong monster on each of the rounds and you should have no problem.


Feeding Frenzy
Successfully feed the dragons 100 times in the dragon-feeding mini-game.

This minigame is accessible when you first reach Viper Manor, at the stables. It's not that hard to do, choose the option to feed the dragons 100 times and with little practice you can do it. If you are having trouble with it, try to use the Slow Speed feature.


Hall of Framer
Collect all 15 types of window frame.

SPECIAL ATTENTION TO FRAMES 5 and 11!

You start, by default, with 3 of them: Arnian Wood, Simple Line and Iron Plate.

4- Name: Shellfish
Location: Arni (Home World) - Search the Element salesgirl's cart to get.

5- Name: Tea for Three
Location: Termina (Another World) - run up the stairs by where the statue polisher is. There is a person standing in the shade of the house to the left (you can’t see this person), and if you talk to him and say you understand him, he'll give the frame. Extra care with this as it seems you cannot get it after returning to Another World after being stuck in Home World.

6- Name: Porre's Furnace
Location: Viper Manor (Another World) - Search behind the Library's desk to find a hidden ladder that leads to a secret room. Talk with the BeachBum to get the frame.

7- Name: Skullduggery
Location: Shadow Forest (Another World) - use the Aroma Pouch to attract the red blob-thing located near the entrance and make it follow you (you understand more about this mechanic if you choose Nikki to guide you to Viper Manor). Bring the red blob to the screen where the big green monster (Quadffid) is blocking the path and feed it to it. It shrinks and gives the frame.

8- Name: Tropical Paradise
Location: Guldove (Another World) – By choosing to not save Kid after she is poisoned, Macha will automatically give it to you when you recruit her.

9- Name: Guldovian Stich
Location: Guldove (Another World) – Choose to save Kid when she is poisoned and keep playing. When Kid gets healed, right after Mel will steal her elements. Choose to search for Mel, even if Kid doesn’t like it. Then go to Termina. When Korcha lends his boat, go and talk to Mel and get the frame.

10- Name: Infared Vision
Location: Fossil Valley (Home World) – Climb the stairs in the first area and search behind the big skull of the big dragon skeleton.

11- Name: Valencian Cloth
Location: Go to Marbule (Another World) with party of only demi-humans characters and talk to the guy in blue at the entrance. Get it while playing as Lynx. You need to do it BEFORE awakening the Black Dragon.

12- Name: Monster's Mouth
Location: S.S. Zelbess (Home World) – When Sneff turn all the party into cats, go upstairs to the kitchen and use the cat door to get inside. Talk to the cat in there.

13- Name: Our Favorite Martian
Location: Sky Dragon's Isle (Home World) - After defeating Starky, reenter the area and talk to the guy who was always running from the monster who walks around the upper part. He'll give you this frame.

14- Name: Snakes & Orbs
Location: Gaia's Navel (Another World) - Get another Ancient Fruit, call the Wingapede in the same area as you already did the process, but now in Another World. Once you get into Gaia’s Navel, find the Prehysteric, a small monster standing alone in a tree. Defeat it and get the frame.

15- Name: Quill and Papyrus
Location: Marbule (Home World) – Return to Marbule after the Chronopolis events to find it settled again. Talk to Toma twice in the Black Dragon’s cave and get the frame.

A Tactical Retreat
Flee from a battle.

As simples as that, get into a fight and choose the option "Run Away" on the menu.


The Long-Lost Element
Obtain Chrono Cross, the rainbow-aligned Element.

After you get both the Tear of Love and the Tear of Hate, and after finishing the events in Chronopolis, go to the Divine Dragon Falls (Another World): it's on a waterfall, northeast of Arni Village. Once there, put both items on the pedestals to get the legendary element Chrono Cross.


The Veteran
Triumph in 500 battles.

Since you will need at least 2 New game+ to get some long term achievements like "The Recruiter" and "Gold Stars for Everyone", you will probably get it sometime.


Gold Stars for Everyone
Obtain 99 level-growth stars.

Same as above, you get a maximum of 48 stars on a single game, you will need at least 2 new game + to get it.


The Ultimate Armor
Create a piece of rainbow armor.

Check The Ultimate Weapon


The Ultimate Weapon
Create a rainbow weapon.

The first step is to recruit Zappa on Termina (Home World), right after you start controlling Lynx and get to Termina for the first time.
To do both the rainbow weapon and rainbow armor, you will need 2 rainbow shells, a rare material in the game. You can find a list with all of them easily on the internet, but i will just point 2 easy rainbow shells for you to get your achievement. For the first one, you can get it on S.S. Zelbess while getting Immoral Compass achievement as well. For the second one, while doing the events on Kid's orphanage, in the first room, the same room where you get Kid's level 7 tech - Hot Shot (check Lucca's Legacy achievement if you wanna make sure), check the clock at the right of the room and a kid will appear. Talk to him and answer "Yes, it's going to be okay".
Then you will need a Master Hammer - first you will need to do the optional quest to save Marbule, right after the events to unmask Fargo on S.S. Zelbess. Then you will need to kill the Black Dragon. After getting back from Serge's body, go to Marbule (Home World) and buy the Master Hammer from a NPC at the same room where you fought the Black Dragon for 10.000G. Then equip the hammer on Zappa and go to Termina (Another World) with him on your team and talk to the other Zappa there. Done, now you can create rainbow gear.
I HIGHLY suggest you to create the Spectral Swallow. The Mastermune is better than Spectral Swallow, since Mastermune do more criticals, but once you start a NG+, you will lose Mastermune but now Spectral Swallow if you have one, which you really help on many NG+ endings.


The Recruiter
Use the Chrono Cross item to recruit all party members.

On a New Game +, by using the Chrono Cross key item (Check The Long-Lost Element achievement) at an area called Temporal Vortex, you can "call back" characters from past games. This is essential, since you make decisions through the game that allow you to recruit some characers and consequently losing others.
The main decisions related to recruiting characters are:
1- your guide at the beggining of the game to invade Viper Manor (Guile or Nikkie or Pierre)
2- Save/not save poisoned Kid.
So you need at least 2 NG+ to get all the characters.
One important note: at some point later at the game, Harle leaves your party permanently, so you need to end a NG+ before she leave.
An example of an optimized way to get this achievement could be (there are more than one):
1- At your normal game, choose Guile as your guide to Viper Manor and also choose to save poisoned Kid (a good walkthrough is essential to get all the optional characters of the game, as there are too many of them);
2- (NG+ 1) Choose Nikki as your guide and choose to not save poisoned Kid. Once you get control of Lynx, recruit Harle and Sprigg and go to defeat the final boss at Opassa Beach, doing also the ending #5 (check The Rebirth of Marbule achievement); (now you will have all the characters from the normal game + the characters from NG+ 1, specially Harle)
3- (NG+ 2) Choose Pierre as your guide to the Manor and follow up the game until after Kid's orphanage events and getting Chrono Cross element as well. Once you are done with this, go to Temporal Vortex by the portal in Hydra Marshes (Home World) and use the Chrono Cross at Sprigg's house. And that's it, you will get all the characters from your past games! :)


The Second Coming
Defeat Dario and reforge Serge's Sea Swallow into the Mastermune.

Dario is a hard optional boss in the game. To most its the hardest one, along with Criosphinx. Anyway, with a good strategy, we can make this fight a lot easier.
But, first of all, to fight Dario you will need an item called Memento Pendant. This item can be obtainable after the events at Hermit's Hideaway (Another World) where a lot of characters join your team to go after Lynx. Once you are done there, head to Viper Manor (Another World). At the room with the elevator and the pillars, head to the east wing that most of the game was blocked by a robot. At the new room, put all the four snake statues at the floor grooves, and a chest will appear (open it from BEHIND). You will get a note to Karsh. Put Karsh into your party, equip him and head to Isle of the Damned (Another World).
Once there, enter the room with many mirrors, go beneath the large spinal column, and use the Garai Keepsake to enter beyond the mirror ther, where you will fight two bosses (just to remember, the boss Peppor has the Forget-me-not-pot accessory for steal, that can really help with the Grand Champion achievement). When you defeat them, you get the Memento Pendant and and can fight Dario.
A good moment to fight Dario is after you kill the White Dragon. Go to the Forbidden Island (Home World). It's a very small island, northeast of the main island. You can identify it as it has small puffs of smoke coming out of it. When you approach it, you won't see a name, but you can press the confirmation button to land there. Put Riddel in your party and also equip her with the Black Plate from the Black Dragon (really hope that you steal it from the Dragon, REALLY help Riddel's chances to survive, as she needs to be in this fight.
Now to the strategy of the fight. If you fight blindly against Dario, specially on your first game, you have really big chances to lose. The main reason is because, besides hitting very hard physically, Dario has high HP and counter all elements. About the elements, the summary is: just use white elements with the character with the Black plate equip; if you don't have the Black Plate, dont use, since Dario will counter with a very powerful black tech. Also don't use black elements, first because black is his element, and second because he will counter with a heal on himself. So you should use just : white elements with someone with BlackPlate, this way his powerful tech will heal the person with the Black Plate; powerful red elements to try to inflict the burn status on him, specially because the counter of red elements is the weakest counter he has.
Bring a lot of healing elements, you will need it. But with patience and knowing what to do and not to do, you can beat him.