Caesar Creek State Park
A popular spot just south of Dayton. This park offers numerous hiking and riding trails, a fossil bed, a (very) small waterfall, and the Pioneer Village - a collection of historical buildings.
Cincinnati Zoo
Smaller and more compact than the Columbus Zoo, the highlight of the Cincinnati Zoo is its Bengal tigers.
Columbus Zoo
I like it better than the Cincinnati Zoo. A more "open" design and lots of different animal exhibits.
Hocking Hills State Park
Ash Cave, Old Man's Cave, three main waterfalls, and plenty of hiking trails make for a wonderful place to visit. Don't forget to stop at the Rock House and Conkle's Hollow. Hocking Hills is my favorite spot in Ohio (so far).
John Bryan State Park
Supposed to be good for wildflowers in the Spring, but I haven't had much luck there. Close to Dayton, the hiking trails can be busy on the weekends. Stop by the old Clifton Mill while you are out there.
Marblehead Lighthouse
Not far from Cedar Point. Can be very crowded on the weekend.
National Museum of the United States Air Force - Dayton
For all you aviation and space flight fans, this museum should be on your must see list. It'll take a day (or two) to get through the four mammoth hangars of exhibts. Arrive early enough and be sure to sign up to tour the Presidential and R&D hangars (you'll need a valid photo ID). Tripods are almost a must given the low light levels. Early morning on Sunday is the best time to visit to cut down on the number of people around the aircraft.