Each trinity ability unlocks at a different point in the story. This achievement can't be unlocked until after you beat Riku at the beginning of Hollow Bastion.
Blue Trinity - Unlocked when you leave Traverse Town at the beginning of the game.
Red - After you clear the first ring of worlds (Traverse Town, Olympus, Wonderland, & Deep Jungle)
Green - After closing the keyhole in Agrabah
Yellow - After beating the Hercules cup
White - After the battle with Riku early into Hollow Bastion world.
Here are all of the trinity locations, listed by world:
Traverse Town
Blue - By the cafe in traverse town (Look for the candles)
Blue - By the world Exit door
Blue - Third district, up the ramp that is behind the water fountain
Blue - Merlin's House by the save point
Green - Accessory Shop (Used to get to synthesis shop) - Thank you Jamone the Afro!
Red - First District, Alleyway
Red - Second District, used to get to the secret waterway (Opened during story progression)
Red - On top of the Gizmo Shop (Opened during story progression)
Yellow - Go outside Merlin's house, its on the back side by some boxes
White - Secret Waterway, where you talked to Leon
Olympus Coliseum
Blue - To the left of gates
Blue - To the right of the gates
Green - When looking at the world exit, go to the right side by the tournament results plaques
Yellow - In the lobby, moved to close Olympus Coliseum keyhole
White - Coliseum Gates, middle of the area
Wonderland
Blue - Lotus Forest - Grove near entrance - grow large by giving the flower a potion and push the square bolder into the water. Shrink again by eating the acorn, then jump up the lily pads and into the hidden area. The trinity spot is in the middle of this small grove.
Blue - Towards the back of the Lotus Forest
Green - Rabbit Hole, behind the save point
Green - Bizarre Room (First section upon entering world) inside the fire place
White - Louts Forest - Go to the back left corner, and climb up. If you turn around, you'll see a doorway behind you on a treetop. Glide over to this, enter, kill all of the heartless, then enter the picture that appears when you turn on the lamps. The trinity will be in that room.
Deep Jungle
Blue - Jane's Camp
Blue - Climbing trees, high on a platform
Green - Arrive at the Treetop - Tunnel save sphere. Go Through the doorway above the save spot, and it's in the center of that area.
White - Waterfall cavern, all the way up top, in the cavern of hearts, in front of the keyhole
Agrabah
Blue - Bazaar on top of a platform
Blue - Cave of Wonders, silent chamber
Red - Cave of Wonders, treasure room
Green - Storage room, by the save sphere
Yellow - Cave of Wonders (Hall) in front of a pottery over. Right past the track where the boulders roll from.
White - Cave of Wonders, Entrance (right in front of the door you come in from the desert)
Monstro
Blue - Mouth, on top of one of the platforms
Blue - Throat, center of the room
Blue - Chamber 5
Green - Mouth, On top of the ship Geppetto's in (right above save sphere) - use glide from the tall platform by the plank
White - Chamber 6, center of room
Halloweentown
Red - Oogie's Manner - Archway near the stream on ground level
White - Moonlit Hill - to the left of the tombstone that uncurls the hill
Atlantica
White - Triton's Palace - center of the purple shell in the area just outside Triton's throne room
Neverland
Green - Ships cabin, unlocked during story progression
Yellow - Ships hold, just outside the room you start the world in
White - Ships deck where you fought Captain Hook, up by the ship's wheel.
Hollow Bastion
Blue - Great Crest - right after the large platform battle
Blue - Dungeon - at the start of the rooms where you were with beast and had to move the wall sections. Beast needs to destroy the wall for you, donald,& goofy to get in
Red - Entrance Hall, 2nd Level in front of statue
Green - Library, second floor
White - Rising Falls (beginning area) halfway up, in the small water basin
If you're a picture person, here's a picture of each location (it's on the PS2, so not the greatest quality):
http://www.supercheats.com/guides/kingdom-hearts/trinity-mar... Thanks to I MulligaN I for the link!
Hope this helps! If you have any questions, please feel free to ask in the comments!