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