Autocross is a great way to begin down the road. It will help teach you basic car control (braking, cornering, acceleration, sliding the car.. locking up the brakes etc) AX events cost about $40 give or take. Many clubs offer loaner helmets (mmm nothing like sharing an old sweaty helmet
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Next up would be a PCA DE event. 1) Make sure you are a PCA member. IF not, pay the $42 and join. 2) Find the local region and see what they have planned. Summer has lots of events. Some relatively close and others far away. Pick one that fits your schedule.
Costs? $200-400 for an event is my experience. Then figure fuel costs ($150?) Hotel if needed ($100?) Car preparation (??) - too many variable to list, but mostly dependant on your cars current condition and how much you can "do yourself" vs sending out to a shop
So.. sign up and have some fun!