There are no restrictions on foreigners buying a car (new or used) in the USA. If you buy it from a dealership they will help you with registration, license plate, and insurance. However, you will probably NOT be able to get a car loan, as suggested by another answer.
You only need an "International Driving License" if your regular license is not in English.. However, it doesn't hurt to have one and may be a good thing if you decide to rent. They are offered through automobile associations and travel agencies.
South African Automobile Association, ph: +27 (0)83 843 22
Most rental companies require age 25. Check with a real travel agent where you live for USA car rental agencies that only require age 21, and they can also get you a price quote for 2 1/2 months.
It is true (as he first answer says) that you can buy a car for a few hundred dollars. However, it would be a junker and would be unlikely to get you form Miami to Florida. Realize that whatever you pay for the car you will get less for it when it's time to sell.
This will give you an idea, and you can research prices of cars online:
http://www.nadaguides.com/
When comparing costs, remember that the agency is responsible for repairs & maintenance of the rental car. In short, if anything goes wrong with a rental car (including an accident) you call the agency and they give you another one. They also pay for things like oil changes and tires. If anything goes wrong with a car you own you pay for it.
Whether you decide to buy or rent make sure you fully insure the vehicle. Getting into an at-fault accident with a poorly insured car is a bad thing, and is worse if you are away from home.
If it was me I would rent, even if the basic cost is slightly higher.
Sleeping in your car may or may not be safe, depending on exactly where. However, many places prohibit sleeping a a car on a public street (it's illegal).
There are national parks, state parks, and private camping areas everywhere that allow camping for very little cost. You can purchase a tent and other basic gear very cheaply at stores like Wal-Mart.
There are lots of camping info resources online and this is just an example:
http://www.camping-usa.com/
http://www.camping-usa.com/campgrounds/
Finally, consider joining SERVAS (www.servas.org), which will give you local contacts in many places even if you don't stay with a host. "Couch Surfer" is apparently similar, but I don't have any personal experience with them.
Good luck!