Here is the full list of all 20 Pony Island achievements. It takes around 4-5 hours to unlock all of the achievements on Windows.
Destroyed the first Core File.
Unlocked Pony Wings.
Destroyed the second Core File.
Killed Louey.
Destroyed the third Core File.
Escaped the arcade.
Learned a secret of your past.
Learned another secret of your past.
Found 7 tickets.
Unlocked Pony Lasers.
Crashed the game.
Found 3 tickets.
Found your first ticket.
Learned a third secret of your past.
Learned every secret of your past.
Found 14 tickets.
Found all 24 tickets.
Found the hidden profile screen.
Found Missingno.
Found an alternate avatar.