Most points within Ohio are easily accessible by auto with longest distances across the state of about 300 miles and less than 5 hours in driving time. For instance, driving from Cleveland-to-Columbus and Columbus-to-Cincinnati via I-71 takes about 2 hours give or take 15 minutes. The larger cities in the state all have public mass transit systems.
Greyhound
connects over 25 of Ohio's cities. The following cities have a dedicated Greyhound station: Akron, Ashtabula, Athens, Cambridge, Canton, Chillicothe, Cincinnati, Cleveland, Columbus, Dayton, Dover, Elyria, Findlay, Kenton, Lima, Mansfield, Maple Heights, Marietta, Marion, Sandusky, Springfield, Toledo, Van Wert, West Salem, Wooster, Youngstown, Zanesville.
By road
Like most of the midwest, highways are long, straight, and flat, and drivers are often going long distances--in other words, a hitchhiker's dream. Be aware that most if not all interstate highways are closed to pedestrians, so you're probably better off trying to catch a ride from an on-ramp. I-80 is the preferred route for those trying to travel across the state east or west.