Yes the Army will pay to transport your car to your next duty station. Also, although it is difficult you can get through BOLC without a car, but I would not recommend it. As an officer and a leader I would expect my LT to show up to any duty station with his/her own transportation.

If you need help, USAA makes it easy to secure a car loan and there are a lot of online websites such as Kelley blue book that will help you buy that first car.