They will fly you to STL for class. After that, you are on your own. They will only positive space you to a base between and after training events. Its a major headache and pain. Be prepared to be hung out to dry repeatedly by the company during your transition. That's how we've all been treated. It sucks, but the only consolation I can offer is that it gets better once you are on the line.

WRT to the car, if you live less than a few hundred miles away, I'd highly recommend take it. It will make your experience much more enjoyable. There is some decent stuff to do in STL if you have a car.
The area around FSI is a crap hole. If they put you up in the Comfort Inn Westport, you have no execuse for not having a good time. PM me if you have any questions about STL or the training process.

You will sign all the paperwork the first day of class when HR shows up and trys to get out of there as quickly as possible. If you have enough questions and make them miss their flight back to IND, you get bonus points.