[Miniata]

Autocross and HPDE/track days are two very different activities. Cost, wear on the car, risk and speeds involved, etc are very different. Both typically require you to commit a full day to the event. Most autocrosses will cost you $25-40 for the day where you will get on average 4-8 runs of less than 60 seconds each and you probably won't go over 60-65 mph. HPDE/tracks days will likely cost you $150-250 per day, there will be multiple cars on course with you, and speeds at a lot of tracks will be in excess of 100 mph.

Find your local SCCA or NASA region, some areas have local non-affiliated clubs that run autocrosses and track days as well, most post their schedules online. For track days/HPDE's you need to preregister and make sure your car meets the tech/safety requirements, and you'll need to buy a Snell rated helmet. For autocrossing you can just show up the morning of the event (make sure you get there with plenty of time to spare), register and pay on site, and at registration, just tell them it is your first time, and they should be able to point you to someone in charge that can help you. Usually there is a novice coordinator that helps the new people. IIRC every autocross I've attended has had loaner helmets available for people who don't have their own.

Focus on having fun, learning car control, get experienced drivers to ride along and give you pointers, and leave your ego at the door. I see a lot of people show up at events in "fast" cars, only to get beat by people in Minis and Neons and Miatas with experienced drivers on better tires, get ticked off, and not come back. Most autocross clubs/regions have driving schools early in the season, and these are always a good idea for new drivers to attend.