There are 2 clubs in your area that I know of.
I'm a member of VMSC. Its a nice friendly club with pleanty of competition if you need it. Our venues are about 1hr to 1.5 hrs from Fredericksburg. Its a little further, but you might like the atmosphere better. I've never been to one of the WDC meets, but they should be closer. If you go to their events, make sure that its open to public, since some of their events are a SCCA members only event.
For instance, if you came out to a meet with VMSC, you would pay 20 dollars, we can lend you a helmet, and your car would be classes STS-N (N being novice). That way, you are competing against all the other new members too. Later, if you like, you could always get your own helmet so you don't have to borrow a club one.