Hmm.
My first car (my 2nd actually, but the 1st one I didnt have to share with my twin brother) was a 96 Honda civic, and I drove the hell out of it. It needed a new tranny and eventuallya new engine (which wasnt my fault, it was because of the tranny install - long story). The only problems I had with it were completely my fault.
Point being, whenever I help people look for reasonable used cars, I always suggest Honda, Toyota, sometimes nissan.
You cant go wrong with a civic or accord, espically if it has been driven by a mom or grandparents. The only maintenance I had to pay for was timing belt and water pump, which should be replaced between 80k-90k inhondas - so if you find a car with that kind of mileage, maybe you can use that as an advantage.
Nothing ever went wrong in that car without my doing (Ie the tranny).
Maintenance is cheap on honda's, and they're not hard to work on, so a local shop should suffice (so you dont have to go to a honda specialist (like vw/audi) or the dealership).
I'm sure you want a safe car as well, because I know most teenagers get into accidents in their first car, the honda had airbags and what not.
I cant speak to toyota's because i've never owned one, but they're pretty solid as well. A few year old camry would be perfect.
As for hyundai, the wife to be (2 weeks!) has one, an 06, and is in love with it (Santa Fe). Im not sure what type of car you're going to get for only 5-7k, and chances are that you're not going to get much out of the 10yr 100k warranty for that price range (75-85k miles might last you a year or 2).
http://www.cars.com/go/search/detail...eadExists=true
Not sure if that car has any warranty left or it might be before the 10yr 100k mile warranty.
Im rambling at this point - I guess I would just suggest looking around at the different dealerships in their "Budget" lots. You might have a hard time right now because the type of car you want to get is very hot (Efficient 4 cylinder), so dealers might not have many and are probably charging too much for them.
It took me nearly 1.5 monts to finally find the car I wanted. GL and Ill be looking out for you!
(PS my parents want to sell my bro's neon, but I wouldnt sell it to you

, I know how he drove it)