Here is the full list of all 26 Buck Up And Drive! achievements.
Finish a run with at least 9001 points
Beat 10 stages in a run
Beat 5 stages in a run
Finished the Endless Road tutorial
Beat 15 stages in a run
Beat 20 stages in a run
Beat 25 stages in a run
Finish a run with at least 25000 points
Finish a run with at least 42069 points
Finish a run with at least 69420 points
Finish a run with at least 100000 points
Finish a run with at least 250000 points
Reach the maximum road length
Beat a Rival
Beat 3 Rivals in a single run
Defeat a police car
Defeat an armored police vehicle
Hit a Police car with their own spiked blue ball
Trigger a Road Rage
Beat a stage in less than 30 seconds
Beat a stage with 0 seconds left, but no crashes
Beat a whole stage in a single drift
Visit the secret test area
Muted the in-game music
Bonk a billboard with a Boost Tackle while in mid-air
Play a match of Shifty Gear