You use your own car. I think this is better, especially for your situation. You want to learn how your car handles. If you are afraid that you will damage your car, that's good, that's how it's going to be on the street (BTW they are as safe as long as you listen to the instructors). These courses are also affordable because they are typically run on a volunteer basis by local clubs. They are kinda like a "public service", so people don't wrap their cars around trees.

The other schools that provide the cars are expensive, because they supply the cars and pay the instructors! They have to maintain them, fix them, replace the tires, etc. They typically also have a clause, where you break it, you pay for it. These are typically racing schools, i.e. real road racing, which does improve skills, but are geared for competition.