Press 'Accelerate' about halfway down and when you get grip and the car lunges forward press it ALL the way down. Doing this will help prevent you from getting stuck even if you start in deep snow or deep sand.
When you use the same car from the beginning of Quick Race mode it is possible to get 2 new Liveries when you reach a certain distance in the game. The first Livery is unlocked when you reach 20 miles (32km). The second Livery is unlocked when you reach 50 miles (81km).