Here is the full list of all 56 Freedom Planet achievements.
Clear the game without losing any lives
Clear Adventure Mode with over 7000 crystal shards
Clear all target rooms in Shang Mu Academy as Carol
Clear all target rooms in Shang Mu Academy as Lilac
Clear all target rooms in Shang Mu Academy as Milla
Collect all 10 cards in Aqua Tunnel
Collect all 10 cards in Battle Glacier
Destroy a Chopper with a single Dragon Boost as Lilac
Collect all 10 cards in Pangu Lagoon
Collect all 10 cards in Thermal Base
Collect all 10 cards in Jade Creek
Collect all 10 cards in Sky Battalion
Defeat an end boss with Lilac's Rising Slash
Collect all 10 cards in Fortune Night
Collect all 10 cards in Relic Maze
Collect all 10 cards in Dragon Valley
Get a boulder to wrap around the screen
Clear all Carol stages
Play the Secret Melody on a piano
Defeat an end boss with Carol's Wild Claw
Clear all Lilac stages
Clear Pangu Lagoon in 8 minutes
Defeat an end boss with Lilac's Dragon Cyclone
Rest on the ledge during Lilac's fight with Spade
Skip the Claw boss in Relic Maze
Clear Dragon Valley in 5 minutes
Skip the switch door in Dragon Valley as Lilac
Collect all 10 cards in Trap Hideout
Let one of your teammates finish off Robopanther
Destroy both of the Mantalith's arms before the fight begins
Clear Trap Hideout in 6 minutes
Clear Final Dreadnought 4 in 8 minutes
Clear all Milla stages
Clear the game in less than 90 minutes
Defeat an end boss with Lilac's Dragon Boost
Clear Thermal Base in 9 minutes
Clear Relic Maze in 8 minutes
Clear Aqua Tunnel in 5 minutes
Clear Sky Battalion in 7 minutes
Trick the Bone Golem in Relic Maze into destroying itself
Clear Final Dreadnought 3 in 6 minutes
Clear Final Dreadnought 2 in 9 minutes
Clear Final Dreadnought 1 in 5 minutes
Destroy Prince Dail's shield before he finishes his monologue
Clear Battle Glacier in 12 minutes
Defeat an end boss with Milla's phantom blocks
Dig into a treasure spot as Milla
Clear Sky Battalion without using Torque's plane
Clear Jade Creek in 8 minutes
Defeat an end boss with Milla's Super Shield Burst
Clear Fortune Night in 8 minutes
Defeat an end boss while riding Carol's motorcycle
Bust some dance moves in Fortune Night
Destroy the Eye Golem in Relic Maze with a push block
Clear Relic Maze as any character
Defeat an end boss with Carol's Wild Kick