To pass to the next level, you must get a Gold star rank in all races and missions. If you do not, you will not pass to the next vehicle type. This goes for both Ford and Chevy.
Upgrading assigned vehicles
When driving an assigned vehicle on a track (for example, Snap It in the Off Road Mode in Chevy), try to buy the car you are going to drive. For example, in Snap It, buy the 2001 Chevy Burggio Concept. Try to upgrade this vehicle.