Autocross is a good way to get your feet wet. And you'll suck at it. We all did when starting out. Open track days and age requirements will vary by the track. If you got to the point of running in a competitive type of event like SCCA, the age requirement is 16. But these are structured classes and only manual transmissions are allowed. My home club track, Waterford Hills, actually made a couple of exceptions based on the racing experience of two 15 year olds who had about 7 years of karting experience (they had their racing license before their driver's license ).

Check out the SCCA website for the club rules for autocross and road racing. As for the road racing, the cheap way to go is Improved Touring (IT). It's an entry level class, but their are no Corvettes. Check it out.