Congratulations and don't worry! If human beings needed doctors and hospitals to be born, we would have been extinct a long long time ago.
If your baby comes au natural, which would be very unlikely for a first baby, she or he will be just fine. DON'T cut the cord! You don't have to. There is absolutely no medical reason baby can't remain attached to the cord for a while as you get yourself to the hospital. The issue there is remaining sterile, and chances are you don't have anything in your car that is sterile, so don't bother. IF it happens (which I'd bet money that it won't) then just deliver the placenta, it will come out on its own, and put it right next to baby in the receiving blanket.
BTW, my son was born at home (planned) and it was safer than being born in a hospital. Read The Thinking Woman's Guide to a Better Birth by Henci Goer for statistical information on hospital intervention procedures. That's what you reallly have to worry about.
Congrats, and don't worry. Your body knows how to have a baby. Your mind doesn't even have to get involved.