Where in VA are you? Summit Point in the north near Charlestown WV (about 90 miles form DC) and Virginia International Raceway (VIR) in near Danville on the VA/Carolina Border bracket VA nicely.
PCA (
www.pca.org) runs great events. There are other groups too like Chin Motorsport, Track Guys, and others.
Most of these clubs run 3-5 30 or so minute sessions, grouped by skill level. Instructors available/required to beginners. Once you get to go fast on the track, you often wonder 1) why you waited so long and 2) just how scary going superfast on the street can be.