If you want to learn to drive really well and turn both ways, I highly recommend SCCA autocross. You "run what ya brung", no special safety equipment needed other than a helmet (Snell rated, not just DOT). It's a great way to dip your toes in, learn a lot and have a lot of fun. And you can drive your race car to work.

I ran a number of daily drivers and junkers... '85 Grand Am, '63 Corvair, '98 Mustang, '82 Skyhawk (the Buick flavor, not the other one). VERY low chance of collision damage of any sort, near zero chance of collision with another car since it's a timed run with (usually) only one car on the course at a time. You can move on to Club and other flavors of racing if you really get into it.
If you want wheel-to-wheel oval track... I dunno. There seem to be all kinds of classes of that, including high school kids flogging the tired shells of old 4-bangers around a dirt track.