I know I'm going to be the only one who suggests this, but...
I got my
97 Saturn SC2 for $4700 may of last year. Pretty good car, its cheap, easy to fix (though, it did break every once and a while, but it seems to be okay now), good gas milage, and looks nice (by my standards). It only had 74k miles, and I've driven about 10k miles on it with only one major problem (tranny messed up, cost about 300 to fix). They don't have loads of horsepower, but they are pretty decent autocross cars (stick to the DOHC "2" series, and don't even bother with the SOHC "1" series). As a matter of fact, the SC2 won last years national championship in its class. Though if you want 4 doors, you can try and find a SL2. I was 17 when I got the car (18 now), pretty much everyone liked it in HS, and was suprised it was so cheap. One of my favorite thing about the car is, whenever you need to get it fixed, Saturns from 91-2002 were pretty much the same car. Repair parts are very easy to find. Don't forget the dent resistant doors. Its great looking at my car, and seeing no dents... because it isn't possible with the plastic exterior!
The SC2/SL2's are cheap, safe (new drivers often crash), plentiful, insurance is low, fairly easy to work on (do most of the work myself), good gas milage, pretty new/good looking (imo anyway

), and they fit your budget perfectly. I'm sure you can find a newer one, with lower miles than I did for your price range.
If you have any questions about these particular cars, don't hestitate to ask

.