Cleveland isn't that bad. I spent a year up there and had a blast. Out of the three metro centers in Ohio I prefer Cleveland.
Rock and Roll HoF is really cool.
Christmas Story House
Go watch the Tribe. There isn't a bad seat in the stadium (bring a light jacket).
Great Lakes Science Center.
Cuyahoga Valley National Forest is really, really cool. It's about 30 minutes south of Cleveland. There's lots of trails and a few waterfalls. Lots to see over in Rocky River, too (about 10 minutes west of Cleveland).
Go visit West Side Market. It's a huge indoor/outdoor market with food from all over.
Go see a play at Playhouse Square
Shaker Heights is an old part of Cleveland. Lots of big, tudor style homes over there.
Food/Drink Recs:
Downtown
Barrio
Lola Bistro (Michael Symon's restaurant)
The Greenhouse Tavern (rooftop bar downtown)
Outside of Downtown (within reasonable driving distance)
Dewey's - Lakewood
Melt Bar and Grilled - Lakewood
Pier W - Lakewood
Great Lakes Brewery - Ohio City
Townhall (great brunch) - Ohio City
Bar Cento - Ohio City
Edisons - Tremont
Tremont Taphouse - Tremont
Cleveland has a fantastic Little Italy (Milla Bella, Mama Santas)
Places to avoid: The stretch of Euclid Ave for about 4 miles east of Cleveland Clinic. Especially after dark.