Here is the full list of all 15 Rising Islands achievements.
Finished a time trial level
Completed the game
Third relic recovered
Second relic recovered
Do a grind
Learned to dash
First relic recovered
Used a spring jump
First powercrystal recovered
Found all collectible crystals in a level
First collectible crystal found
Ran on a wall
Experienced failure
Completed the tutorial
Found all collectibles in the game