Here is the full list of all 20 Drive!Drive!Drive! achievements.
You completed Campaign mode!
You jumped more than 300m.
You drifted more than 400m.
You smashed into one of your own cars. From another track, obviously.
You smashed into an opponent from another track.
You drove around a corkscrew.
You came 1st in all tracks in a race.
You drove around a loop the loop.
You managed to turn your car upside down, many congratulations.
You obtained a medal of the golden variety.
You drifted more than 200m.
You jumped more than 100m.
You smashed into one of your own cars from another track whilst in the air.
You smashed into an opponent from another track in mid-air.
You smashed into an opponent in mid-air.
You were taken for a ride. This time by a car rather than another human being.
You gave an opponent a piggy back.
You made contact with a wall. Well done.
Yalopolooza! You smashed an opponent up.
You completed Offensive mode! That's pretty hardcore of you.