I was in Columbus a couple of weeks ago for work and stayed over for the weekend and made a trip up to Cleveland. At that point, everything I knew about Ohio came from watching WKRP, Major League, and COPS, so I was curious what the place was like.
In Cleveland I went to the R&R HOF. Was somewhat disappointed, but then I'm not a huge rock buff. Next door is another big museum, science I think. There's also a submarine you can tour next door, and right behind the HOF is a boat tour you can take of the lake. I wish I'd done the boat tour just to see what it was like.
I also stopped off in Canton for the Pro Football HOF. Outside of the hall with all the busts, I was disappointed in the place. Granted they were doing some construction, but it won't be finished by the time of your visit.
In Columbus I'd recommend a minor league baseball game. The stadium there has received from great reviews and does look like a pretty cool place. I went to the C'bus zoo and was pretty impressed (got to see some giant tortoises getting it on, which is something you don't see every day!).
I'm sure there's a lot more to the area than I got to experience, but thought I'd share what little insight I had.