This is a difficult question to answer because there are different types of costs associated with going racing. There are the administrative costs—membership, license, and for some an SCCA medical approved by a doctor—which are a few hundred dollars. Next there are event entry fees which range from $400-$700 for an event depending on the track. Following that, if you own your own car, you need the car and most likely a truck and trailer and running costs for each event. If you rent a car from a Prep Shop, you’ll still need to get yourself to the track and of course decide where to stay (hotel or camping) and what to eat (packing food or going to restaurants).
At the very economical end of the spectrum, budget $15-$20,000 for your own car/truck/trailer, and at least $1,500/event to enter and run the car. Vehicle rental typically costs between $2,000-$5,000/weekend but you still need the entry fee and your personal transport. Of course, you can spend a lot more depending on your choices of race car, tow vehicle, trailer, lodging, etc.