Here is the full list of all 36 Harmony's Odyssey achievements. It takes around 6-8 hours to unlock all of the achievements on Windows.
Complete all Puzzle levels
Complete all arcade mode levels
Complete all Cat's Delivery Service levels
Complete all levels on Hard difficulty
Complete the campaign
Recover cat's memories
Recover all missing cat parts
Unlock Werestorms Attack mini game
Help Dragonmice escape
Spend some time behind bars
Make Odin really angry
Get Cyclops a winter treat
Discover Winter Resort
Unlock Detective Mummy Part 3 mini game
Unlock Dungeons'n'Mice mini game
Finished a level in over 25 minutes
Get Detective Mummy angry
Fly through the mythical ocean
Unlock Detective Mummy mini game
Spend your first earned star
Visit the Audio Room in Home Hub
Get Cyclops mad
Discover the Airport
Complete all Bull's Football League levels
Unlock Bull's Football League mini game
Get Minotaur mad
Discover Bull's Football Field
Do cats even like sausages?
Unlock Cat's Delivery Service mini game
Discover the family collection
Get Cerberus mad
Take a look at the city rogues
Discover Olymp Estate
Discover the arcade room
Unlock Detective Mummy Part 2 mini game
Discover Harmony's home