You should read this page here:
HPDE: High Performance Driving Event - National Auto Sport Association - nasaproracing.com
Also check out Chin Motorsports, if not to join, at least read the info on their site (especially the message board), TONS of info there.
Chin MotorSports - More Tracks, More Track Time!
The amount of events before you no longer need an instructor is completely based on your skill and comfort level. They decide when you are ready, and you decide after that if you want to go solo or not. It's also usually on a per-track basis. If you've done Road Atlanta 10+ times and you are cleared solo, that doesn't mean if you go to a new track you'll just be solo, you have to qualify again usually. There are exceptions to this though.
Oh, and it isn't "racing", all of the groups that do HPDE's are always sure to point this out, it's very important.