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

First Wedge (Bronze): Win your first wedge (Classic).
Full Circle (Bronze): Fill your puck for the first time (Classic).
Classic Champ (Bronze): Win a game of Classic Trivial Pursuit.
On a Roll (Bronze): String together 5 correct answers (Classic).
Keep on Rolling (Bronze): Land on a roll again space twice in a row (Classic).
Fast Worker Entertainment (Bronze): Acquire a Entertainment wedge on your first attempt (Classic).
Fast Worker Geography (Bronze): Acquire a Geography wedge on your first attempt (Classic).
Fast Worker Science and Nature (Bronze): Acquire a Science & Nature wedge on your first attempt (Classic).
Fast Worker History (Bronze): Acquire a History wedge on your first attempt (Classic).
Fast Worker Sport and Leisure (Bronze): Acquire a Sport & Leisure wedge on your first attempt (Classic).
Fast Worker Art and Literature (Bronze): Acquire a Art & Literature wedge on your first attempt (Classic).
Wedge Off Winner (Bronze): Win a Wedge Off (F&F).
Lucky Break (Bronze): Win a wedge by betting (F&F).
Gambling Glory (Bronze): Place 5 correct bets (F&F).
Risky Business (Bronze): Use a Bonus Event for the first time (F&F).
Wedge HQ Winner (Bronze): Win a wedge using the Wedge HQ space (F&F).
Bet Your Bottom Dollar (Bronze): Win all your wedges by betting only (F&F).
Winner (Bronze): Win a game of Facts & Friends.
Cleared the Board (Bronze): Clear the board.
First Objective (Bronze): Complete your first objective (CTB).
Objective Success (Bronze): Complete 5 objectives (CTB).
Objective Champion (Bronze): Complete 10 objectives (CTB).
Objective Master (Bronze): Complete all objectives (CTB).
Maxed Out (Bronze): Achieve a x10 multiplier on every category (CTB).
Personal Best (Bronze): Break your own high score (CTB).
Unstoppable (Silver): String together 10 correct answers (Classic).
Gambling God (Silver): Place 10 correct bets (F&F).
Not a Scratch (Silver): Win the game without losing a life in the end game phase (F&F).
Domination (Silver): Win a 4 player game of Facts & Friends without losing a life.
Flawless Entertainment (Silver): Answer all Entertainment questions correctly in a single game (F&F).
Flawless Art and Literature (Silver): Answer all Art & Literature questions correctly in a single game (F&F).
Flawless Science and Nature (Silver): Answer all Science & Nature questions correctly in a single game (F&F).
Flawless Sport and Leisure (Silver): Answer all Sport & Leisure questions correctly in a single game (F&F).
Flawless Geography (Silver): Answer all Geography questions correctly in a single game (F&F).
Flawless History (Silver): Answer all History questions correctly in a single game (F&F).
Straight To The Point (Silver): Clear the board using only the Wedge HQ spaces.
Knowledge is Invaluable (Silver): Break 60,000 points (CTB).
You Were A Blur (Silver): Clear the board in under 10 minutes.
Wedge Domination (Gold): Win all the wedges in a single game (F&F).
You Know Too Much (Gold): Clear the board without getting a question wrong.
Platinum Trophy (Platinum): Earn all other trophies.

Additionally, there are four secret trophies:

You don't know everything (Bronze): Get a question wrong (Classic).
Maybe you don't know anything? (Bronze): Get 5 questions wrong in a row (Classic).
Better luck next time (Bronze): Be the first person to lose all their lives in the end game phase (F&F).
Come on! You're not even trying! (Bronze): Get 10 questions wrong in a row (Classic).