SCCA Sports Car Club of America
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Those are the two biggest organizers of races in the US. For NASA I would look at the time attack or HPDE open track events. For SCCA, look up their Solo 1 (autocross). SCCA Solo 1 would be your best bet, since you can drive up in almost any car, without any experience, and the entry fee is about $20. They can even hook you up with a helmet for the day.