Complete one of the following goals to unlock the corresponding bonus car:

Muscle car
Find 4 keys in any track. Then at the transmission selection screen, press Down.

Hot rod
Find 8 more keys in any of the other tracks. Then at the transmission selection screen, press Down.

City bus
Find all 16 keys. Then at the transmission selection screen, press Down.

Police car
Complete tournament mode. Then at the transmission selection screen, press Down.

Taxi cab
Complete circut mode. Then at the transmission selection screen, press Down.