In reply to David Williams :
Definitely no need for a new car for the first few times. It's probably better if you don't. Figure out if you even enjoy it first, and what you enjoy most. Some people love high horsepower, some people prefer light and quick. I've driven both and as someone with limited natural talent, I've found I like a more modestly powered lighter car that I can really push to its limits vs. a monster that I can't keep up with. (not full on miata modest, but not a z06 either, like e46 3 series ish)
I'd definitely recommend a full brake fluid flush with good quality high temp fluid (ate type 200 is probably the cheapest fluid that I'd be comfortable with - not all "dot 4" fluids are the same) and make sure you have a decent set of pads. It's scary to suddenly lose brakes, especially for a novice. In fairness my "home track" is exceptionally hard on brakes so YMMV.
Just go out with what you have, listen to your instructor(hopefully you get a good one), stay relaxed and have fun. If the bug bites you, and it probably will, you'll know what you want. (turns out I'm a BMW guy, I didn't know that going in, lol)
As for autocross, and I know I'll get spat on for saying this, I tried it and found it incredibly boring. Standing around picking up cones all day for 3 minutes of driving is not my idea of fun. That said, some people love it, and more power to them - you might be one of them. My point is, If you try that first, and don't get what the fuss is about, don't let that discourage you from a track day(and vice versa... I guess, I'm biased though). Two totally different animals.
Lastly, I'd definitely recommend going with an instructor vs. just showing up at an SCCA track night and trying to figure it out yourself. There is a surprising amount to learn, and better to learn the "right" way first than picking up bad habits before even getting in to it...
Just sign up and go already!!!!
p.s. I think you'll probably find that you'll want a dedicated track "beater" that you aren't afraid of breaking, and willing to take some weight out of(gutting) rather than taking a nice new Camaro out there and worrying about scuffing the paint...