Here is the full list of all 30 Rainbow Cotton achievements.
Beat Stage 1
Beat Stage 2
Beat Stage 3
Beat Stage 4
Beat Stage 5
Defeat Tweed
Beat the final Battle
Beat the game
Get first place on the Leaderboard
Use a magic Power
Defeat a Boss without being hit
Fly through every possible path
Use the same magic type three times in a row
Reach shot level 2
Reach max shot level
Defeat a mini boss
Defeat all mini bosses
Beat a Stage without using magic spells
Beat a stage in 2 Player Mode
Beat the game in Retro Mode
Beat the game without dying
Beat the game on highest difficulty
Free a fairy
Free all fairies
Use a fairy Attack
Sacrifice a fairy to cast stronger magic
Defeat a Boss in 25 Seconds
Defeat a boss by him fleeing
Collect no Tea during Teatime
Collect 10 Teas during one Teatime