'Amish Country' makes for a great day trip, though don't go on a Sunday, as you'll find that practically everything is closed! You can drive an hour or two, in several different directions from Cleveland (on Interstate 77 or 271) and find yourself in another world... literally. It's farm country where you'll find Amish people living. This religious faction lead a very primitive lifestyle - with no electricity, no vehicles (there transportation is a horse-drawn buggy) You'll find yourself in beautiful green, grassy surroundings, with craft shops and cheese-houses, where the Amish sell their wares.... and if you make it there, you HAVE to sample the Amish wine. It is unbelievably cheap and really rather good.
Coventry....Well this is just an interesting street with all sorts of little shops, cafe's and bars. There is even a great art movie theatre. I also recommend going into 'La cave da vin' a cozy little wine bar. And in the evening try the 'Night Kitchen' at the Dobama theatre for some interesting theatre productions...
Cleveland has a lot of different cultures and atmospheres. Talk to people and find out where places like Little Italy, Tremont and Ohio City (but don't walk around especially at night!), Coventry, University Circle, and Shaker Sqaure are. Most of these are quaint places which have lots of restuarants and shop. The museums are by University circle.
Little Italy is a yummy place to spend an afternoon or evening... Lots of good food. Parking may be a problem so be prepared. But, there are tons of little art galleries to meander in and out of. Don't forget Presti's for breakfast...great pastries.