One place in the Carolinas is
http://www.carolinamotorsportspark.com/ . I am a founding member of the club, have been racing or "spirited driving" here for about 12 years. I have a few track cars, or cars that I track. You can go full stock, or full race, doesn't matter. I even saw a Ford Explorer once.... It can be expensive if the bug bites, I have been engaged with PCA, BMWCCA, NASA, and Rezoom Motorsports all over the Southeast, they all do great events from beginners to club racing. An easy way to try is to do a single day at CMP above, I think it is about $150 per day. You can even take a rental car if you want to.
Most tracks make you sign waivers, so you are on your own if you end up bending the car. I have had minor disagreements with tire walls in my past, it will happen if you track a lot and push the limits of talent against adhesion. Scariest track around here is Road Atlanta, awesome elevation changes, but a lot of concrete wall.
Another option could be to build or buy a Lemons car. I have done this a few times. 24 Hours of Lemons, cars must cost less than $500, need full cages and no limits to safety equipment investment. I have worked on 5 cars for this, 1 Kia, 1 BMW E36, 2 Porsche 944s, 1 Mazda 626. The Kia was indestructible, the 944s the most unreliable. If the club wants to enter, typically need 4-6 drivers on a team as the race is long. could cost around $300-500 initially for fees and car. I am in if there are enough people to do it. We need someone with a trailer and a place to store it when not racing it. Last one I worked with was with Nick Short of Competition Race Parts (he builds pro race cars), and it was a hell of a weekend.
Lemons typically comes to CMP in the Spring, so there is time.
https://www.24hoursoflemons.com/events-results/article/145-southern-discomfort