There is a public train station at the airport. Here is the link to the scheduled of trains and buses.
http://www.riderta.com/routes
Sometimes there will be a walk from a train to a bus connection of 1/3-1/2 mile. It might be easier to rent a car from airport and get started on your trip.
Some things to see in Cleveland.
Downtown, along east 9th st: Great Lakes Science Center, Rock and Roll Hall of Fame, Goodtime 3(Think it is 3) lake cruise, Mather steam ship and the USS Cod submarine are all in one area.
If you take the rapid train from the airport to Tower City you can shop and go up in the observation deck of one of the tallest buildings in the city
West side market is a diverse market with produce, meat and prepared foods. Nearby is the museum/house exterior for the Christmas story movie.
The federal reserve has a branch in Cleveland. I believe it still offers tours but check the website.
Their is a Art museum, a natural history museum and an antique auto museum and if memory serves me they are all near each other.
Lots of green space parks and beaches along the Lake Erie water front. If you want to drive west of Cleveland there is a nice boat ride to Put-In-Bay, a quaint island community.
South of the city in the town of bath is a replica working 1860 homestead. Their is also a working steam train that goes through the countryside leaving from the south side of the city. Cuyahoga valley Train I believe.
There are many other historical sites like the presidential homes of several presidents including Garfield, as well as old tow paths with locks.
The drive to rochester will take you through Erie PA so you can look at stuff to do there. Once you cross into New York there are several wineries in the Chautauqua area. You will pass through Buffalo as well.
I would also strongly consider make a short detour to Niagara Falls if you haven't been there. You can stay on the American side or cross into Canada (I believe you need a passport or passport card now to cross over) for the better view. The Maid of the Mist is a great experience where you take a boat near the bottom of the falls. Up the hill from the falls is pretty touristy but the falls themselves are neat as is the drive along the niagara river up to the falls.