Here is the full list of all 31 Frogun achievements.
Win all races
Beat Hexy
Take a photo of the scenery
Beat Scorchy
Beat Wintry
Pose for the camera
Beat Kicky
Finish a level without damaging any enemy
Get a Max. Health Up
Beat Prickly
Buy a hat
Win a race
Fall on water
Finish a level with all of the emblem pieces
Finish a level under the time limit
Get all coins in a level
Finish a level without dying
Finish a level with both eyemeralds
Finish a level having found the notes
Finish a level with the obsidian skull
Beat Beelzebub
Get all of the complete emblems
Buy all hats
Buy all gallery images
Complete the bestiary
Get all health upgrades
Beat the Boss Rush
Beat the Brutal Rush
Own the Kiwi and Melon hats
Defeat a boss cosplaying as them
Hit Jake with something during a race