Here is the full list of all 16 Toybox Turbos achievements. It takes around 2-3 hours to unlock all of the achievements on Windows.
Played a local match with more than 1 player
Unlocked a new cup
Completed an event on every track in single player
Achieved a 3 star rating for every event in a single cup
Achieved a 3 star rating for every event in the game
Achieved 3 stars in a Countdown event
Achieved 3 stars in an Escape event
Achieved 3 stars in a Classic race
Achieved 3 stars in a Time Trial event
Achieved 3 stars in an Overtake event
Purchased any vehicle
Collected all vehicles
Repainted any vehicle with a custom colour
Beat all boss battles
Earned 1,000,000 coins
Played an online match against any opponents