Here is the full list of all 16 Retro Kart Rush achievements.
Open the game once.
Unlock the first secret character
Unlock the second secret character
Unlock the third secret character
Get first place in Donut Cup!
Get first place in Coconut Cup!
Get first place in Coffee Cup!
Get first place in Chocolate Cup!
Get first place in Milk Shake Cup!
Get first place in Ice Cream Cup
Get first place in Random Cup!
Get first place in Master Cup!
Get first place in any cup on the insane difficulty
Get all 32 stars
Play time trials
Play the game with 3 other friends