One place would be a NASA competition licensing school, which are held at VIR several times a year. You will need a logbooked racecar (which you can rent) and will need to have graduated through the HPDE program first (which takes 10-20 weekends on average):

http://www.nasaracing.net/hpde.htm

Once you gain a license, the closest two tracks you could race on are:

http://www.summitpoint-raceway.com (outside Winchester VA)

and

http://www.virclub.com (outside Danville VA)

Enter "Summit Point" or "VIR" into Youtube and soforth... and you can see plenty of in-car videos.

You can also do the HPDE's and/or time trials mentioned at the link above at these same facilities. Often during the race weekends, with NASA at least.

It doesn't HAVE to be autocross. Although a lot of people recommend starting there first.

The info at http://www.nasaracing.net will get you started...

Hope it helps,

Jon