The 13 vehicle classes you need to win races with are:
Dirt Bike
ATV
Buggy
Rally Car
Racing Truck
Mudplugger
Big Rig
Monster Truck
Superbike
Chopper
Supermini
Super Car
Muscle Car.
You will pick up 11 of these 13 by playing through the Rookie chapter of the Festival assuming you win the events and will only need Chopper and Super Car.
However if you wish to get these quickly you can do a 1 lap quick race on an easy track like 'Ship Shape'