Here is the full list of all 16 To the Core achievements.
Finish a normal (random) game.
Finish a game without taking damage.
Finish a Weekly Challenge.
In a Weekly Challenge, reach the top 10 on the leaderboard.
Destroy a Core atop a tower.
Destroy a Centurion.
Jump on 5 different Scroombas without touching the ground.
Hit four enemies with a single ground slam.
Hit three Scroombas with a single Boomba detonation.
Destroy 500 robots across all games.
Dash-push a Scroomba into the secret goal.
Jump on a Scroomba as it's dashing towards you.
Spend a Power Core at a shop.
Obtain five upgrades in a single game.
Finish a game with at least five unspent Power Cores.
Break 500 crates across all games.