Julie:
I too have looked into this a little bit. I don't think you can insure your car. Therefore, if anything happens to it while you're at the track you're SOL.
You don't need to buy anything for your car. Of course, you could make some mods that would improve your car for the track. Better tires, brakes, roll cage, rims etc.
You can run your car just as it's set up now, however. I would also suggest saving some money for the hobby. Depending on how hard (or long) you're running you may have to replace your brakes, oil, and tires after a track session or two. Other things like a clutch, etc., may go as well. Be prepared.
The only thing keeping me from the track is the inability to have a 5-10-15K track fund.

The other choice would be to run the track at a school for a weekend (probably your wise choice) or buy something specifically for the track - like a late model Miata or so (or 2004 ZO6 - my goal)
Autocross is probably the safest option - all you'll really need to replace are you tires. The strip is good too, but that doesn't seem your style.
I've only gone to the dragstrip a couple of times with my old Cobalt SS.