Here is the full list of all 14 Rise of the Slime achievements.
Finish your first run
Play 10 cards in one turn
Teleport during combat
Get your first pet
Change your crown to something more fancy...
Check out what is inside those pots
Push a button
Set up camp, grill some snacks, get some health
Travel trough a portal
Acquire a core mutation
Start a new run with unspent 3000 GP
Perform an execution
Win cash from the lucky wheel
Enter your second loop of the game