Games like this can be confusing, especially the first few times you play them! To start a car in DAYZ, you’ll need to hold on the right trigger button. The car should begin driving.
If you want to shift gears, you can press B to shift up and X to shift down. Some cars will have different controls depending on the model of the vehicle, however, so if you’re having trouble, make sure to look up the specific vehicle that you build to make sure you’re using the right controls.