The stabilitrak light and engine light comes on when I start the car.

Hello, your vehicle is new enough that I would suggest guidance by your local dealership, as hopefully you still have some factory warranty available to you. The Stabilitrak light coming on indicates a fault within the vehicle stability control system, and often the anti-lock brake system which utilizes some of the same components. Because you have a "violent jerk" from the vehicle when shifting to reverse, it could also have something to do with sensors that influence transmission operation. Testing is required to find out what is happening, and come up with a solution to resolve the concern.