The large cities on the east coast (Boston, Philadelphia, Washington) are very easy to get to via public transportation from NY. The train is quicker but the bus is less expensive.Of course it s possible to get to Ohio, but I believe your only option would be the bus.If you re looking to get prices and/or time estimates, I suggest you look at greyhound.com (for the bus) and amtrak.com (for the train).
To go from the East to Ohio there is of course bus options. But depending upon what city you are leaving from Southwest Airlines might be cheaper.For 10 each way you can take a bus to Philadelphia, for 20-30 you can take a bus to Wash D.C., probably the same to Boston.