Unless you can get your parents to rent the car, I doubt you'll be able to rent a car yourself, regardless. I think most places won't rent to anyone under 21~25.
You might look for a $500~1000 beater car/truck with stick to practice on. This way you aren't out too much money when you're ready for another car. Get a job somewhere and earn some money to fix up or save to buy a newer car while the beater still runs.
F-150 with the 300ci V6 and stick shift, Ford Ranger 4cyl stick shift, Pontiac sunfire, chevy cavalier, Ford Escort, Hyundai Accent are good choices. Any Honda or Toyota will likely be TOO beat up in that price range. Those vehicles should have plenty of parts out there to fix them up when something breaks on it.