Successfully complete one of the following tasks to get a trophy:

Out of this World (Bronze): Awarded for casting a Gate spell and traveling to Ni no Kuni.
Big Spender (Silver): Awarded for successfully stealing 50 times.
Artful Dodger (Bronze): Awarded for successfully stealing 50 times.
Soul Patrol (Bronze): Awarded for collecting 2,000 soul orbs.
Little Battler Experience (Silver): Awarded for winning 1,000 battles.
Imajinn-ary Friend (Bronze): Awarded for maximizing an Imajinn's affection level.
Pedigree Breeder (Bronze): Awarded for raising an Imajinn to its maximum limit.
Viva the Evolution! (Bronze): Awarded for completing 10 Imajinn evolutions.
Imajinn-native (Bronze): Awarded for finding 20 types of Imajinn.
Boy Scout (Bronze): Awarded for carrying out 15 different favors.
New Sheriff in Town (Bronze): Awarded for hunting down 10 different bounties.
Pop Pop Fizz Fizz (Bronze): Awarded for alchemizing 10 different items.
Mad Scientist (Gold): Awarded for alchemizing 120 different items.
Treasure Hunter (Silver): Awarded for finding all hidden treasure chests.
Globetrotter (Silver): Awarded for setting foot in all secluded regions.
Raising the Stakes (Bronze): Awarded for collecting your first prize at the casino.
Magic Master (Silver): Awarded for learning all known spells.
Prima Donna (Bronze): Awarded for learning all songs.
Gunslinger (Bronze): Awarded for learning all kinds of shots.

Additionally, there are 15 secret trophies:

Wizardry Whiz (Platinum): Awarded for becoming a master magician. Congratulations! You truly are a whiz at wizardry!
En Guardian! (Bronze): Awarded for defeating the forest guardian and bringing him back to his senses.
Anchors Aweigh (Bronze): Awarded for obtaining the good ship Queen Cowra.
Fly the Friendly Skies (Bronze): Awarded for befriending Kuro.
Wonder Wand (Bronze): Awarded for restoring the legendary wand 'Gradion' to its former glory.
Dark Djinn Downthrower (Bronze): Awarded for defeating the Dark Djinn.
White Witch Walloper (Silver): Awarded for saving Ni no Kuni by defeating the Queen of the White Ash.
King of the World (Gold): Awarded for defeating the guardian of the world.
Imajinn-nation (Gold): Awarded for finding 250 types of Imajinn.
Humanitarian of the Year (Silver): Awarded for carrying out 60 different favors.
Bounty Hunter (Silver): Awarded for hunting down 40 different bounties.
Super Hero (Gold): Awarded for collecting all the hero licenses.
High Roller (Silver): Awarded for collecting all of the tickets for the casino's theater.
Pandora League Challenger (Bronze): Awarded for winning your first bout at the Pandora League.
Pandora League Champion (Gold): Awarded for becoming the reigning champion of the Pandora League.