I've purchased a car in New York and drove it back to Texas before and it wasn't that hard. Some things:
1. Make sure the title is in order so you don't have any problems when you bring it home.
2. You will have to get it inspected when you get back to Ohio
3. Also you will have to get an Ohio registration.
4. Profit.
Really, for me it was an easy experience, my main thing to look out for is the title being filled out right and clean to take it home.