Here is the full list of all 14 Rise of the Slime trophies - there are 1 bronze, 2 silver, 10 gold and 1 platinum.
Unlock all other trophies
Finish your first run
Start a new run with unspent 3000 GP
Play 10 cards in one turn
Change your crown to something more fancy...
Perform an execution
Win cash from the lucky wheel
Enter your second loop of the game
Get your first pet
Teleport during combat
Break 10 pots
Push a button
Set up camp, grill some snacks, get some health
Travel trough a portal