There are usually many groups who do them. SCCA events are usually much more competition oriented events than just open track. NASA does their HPDE events, where you do go out with instructors, and many marque-specific clubs have their own days (BMW, Mustang, Corvette, etc.) but are usually pretty open to allowing anyone in as long as they aren't dangerous and can help them offset the cost of the trackday. Many tracks have their own open days as well.
At Mid-American Motorplex
here in Iowa you can pay $300 a year, and $40 per day, and they have open track days all over the place, including just about every wednesday from early spring to late fall.