Here is the full list of all 22 Lunistice achievements.
Find all Box Buddies in all of Hana's levels
Get all S-Ranks in all levels as Hana
Collect all Cranes as Hana
Ruin the cat slimes' vacation
Get all S-Ranks in all levels as Toukie
Land on the big Jerry's head
Collect all Baubles as Toukie
Beat the game as Toukie
Get all S-Ranks in all levels as Toree
Collect all Stars as Toree
Beat the game as Toree
Remove all crystals in Stage 3 - Act 1
They were kinda asking for it
End Lunistice
Beat Stage 7
Beat Stage 6
Beat Stage 5
Beat Stage 4
Bonk.
Beat Stage 3
Beat Stage 2
Beat Stage 1