Here is the full list of all 21 Don't Look Down achievements.
Destroy your first robot.
Find your first bonus room.
Grab your first jetpack.
Grab your first armor.
Shoot your first burn ball.
Make your first purchase.
Beat all Zone 1 levels.
Smash 50 robots.
Grab 50 powerups.
Find 10 bonus rooms.
Find 20 bonus rooms.
Beat all Zone 2 levels.
Beat all Zone 3 levels.
Destroy 10 enemies with burn balls.
Destroy 30 enemies with burn balls.
Purchase all cosmetics.
Spare Gearheart
Smash Gearheart
Beat the final level.
Beat all Zone 4 levels.
Beat all bonus rooms.