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

First in our Hearts (Bronze): Cleared Generation 1.
Second to None (Bronze): Cleared Generation 2.
Fleet of Foot (Bronze): Raise your AGI over 50!
Stout of Heart (Bronze): Raise your VIT over 50!
Strong of Body (Bronze): Raise your STR over 50!
Sound of Mind (Bronze): Raise your INT over 50!
Stroke of Luck (Bronze): Raise your LUK over 50!
It's a Trap! (Bronze): Captured a monster.
Ch-Ch-Ch-Ch-Changes (Bronze): Converted an item.
It Happens to a Lot of Guys (Bronze): Nope, he did it again.
It's the Only Way to be Sure (Bronze): Defeated an opponent with an Overkill.
Better Safe Than Sorry (Silver): Overkilled 500 enemies.
It's My Specialty (Bronze): Successfully executed a Special Art.
Monstrous (Bronze): Combined monsters.
Nirvana...? (Bronze): Created a marionette.
Stay Classy, Kraltarla (Bronze): Change your class. This takes a special item...
Til You Drop (Bronze): Went shopping 500 times.
Certified Pre-Owned (Bronze): Sold 1000 items.
Fight in the Shade (Silver): Defeated 10,000 enemies.
Love is a Battlefield (Silver): Engaged in 2,500 battles.
Real Soviet Damage (Silver): Dealt 10,000,000 points of damage.
You're Already Dead! (Silver): Try and get a hit count of 650.
Spoils of War (Bronze): Earned 25,000,000 G.
Enhance! Enhance! (Bronze): Earned 25,000,000 EP.
Point Technician (Bronze): Earned 10,000 TP.
You're En Good Hance (Silver): Enhanced items over 5,000 times.
Homeowner (Bronze): Got your first title!
Landlord (Silver): You have more than 100 titles.
Monarch (Gold): You have all the titles. Good work!
Profilific (Bronze): Use all profiles.
David Attenborough (Silver): Completed the monster picture book.

Additionally, there are 13 secret trophies:

Platinum (Platinum): You've acquired all the trophies.
Truth and Consequences (Gold): Saw the true ending.
The Important Bits, at Least (Silver): Cleared the digest route.
Time and Again (Bronze): Successfully executed a distinctive Special Art.
The Combine (Silver): Combine monsters 100 times.
He Chose...Poorly (Bronze): Changed Ellis's appearance.
Son of a Beach (Silver): You have visited the beach.
Hot Springs Eternal (Silver): You have visited the hot springs.
I Know (Silver): You reached the end of a generation with at least one woman completely in love with you.
Don't Get Cocky! (Silver): You've defeated the monstrous Decimal.
They Mostly Come Out at Night (Silver): Defeated the four types of hidden monsters.
We Meet at Last (Silver): Encountered the worst enemy.
Dungeon Master (Silver): Cleared all the hidden dungeons.