Here is the full list of all 29 Luna and the Moonling achievements.
Touch a Fallen Star for the first time
Meet the Moonling for the first time
Luna and the Moonling both faint at the same time
The non-controlled character faints
Collect 10 Fallen Stars from step goals
Collect 10 Fallen Stars from time goals
Collect 50 Fallen Stars from step goals
Collect 50 Fallen Stars from time goals
Unlock the second level page
Unlock the third level page
Unlock the fourth level page
Beat all the Moon Isles levels
Get the Magic Potion on the final level
Get the Tome of Knowledge on the final level
Get all Stars on the Moon Isles Quest
Get all Stars on the Desert Barrens Quest
Cast a Super Moon Shot
Causing Spitter Shot to kill a Boom Fly
Barely avoiding a Spitter Shot fireball
Use a Boom Fly to destroy two other Boom Flys at once
Switch between Luna and the Moonling 1000 times
Moonling pushes 1000 things
Cast 1000 Moon Shots
Walk 10,000 tiles
Faint in the water 50 times
Get hit by a fireball 50 times
Destroy 50 Spinsters with a Super Moon Shot
Ride a Furttle 500 tiles