While there is no single set of generic 'must see' attractions between any given pair of major cities, there are a few worthwhile rules-of-thumb in planning a RoadTrip. One of them is to break up long drives with relatively frequent short stops at places of interest. Another is to take different routes out and back on a round trip. You seem to have both items in your planning. So a few suggestions to help you with that planning.
The shortest route basically follows the interstates, I-270/I-70/I-76/I-80, and that includes the Pennsylvania Turnpike (tunnels!) and the Ohio Turnpike. There will be tolls but they are not prohibitive. If you want to visit Pittsburgh and spend some time there, then this is the route you should take for that leg, since it goes right by that city and you can use I-376/I-79 to get into it. Also on this route are Antietem, Fort Frederick, a unit of the Chesapeake and Ohio Canal National Historical Park, and Tinkers Creek State Park.
For your other route, you might want to take a more 'back roads' approach. This would also give your daughter a chance to act as navigator to help you find your way because the route I'm going to suggest is a bit complicated, and while I'm going to describe it westbound you can, of course take either route in either direction. For this 'back roads' route, you'd leave the DC area on VA-7 through Leesburg to Winchester. From there, take US-522 west past Cross Junction and take VA-127/WV-127 (Has your daughter been to West Virginia?) and then WV-29/WV-9 north to Paw Paw, crossing the Potomac into Maryland and a chance to visit a different portion of the Chesapeake and Ohio Canal\. On the Maryland side of the river, MD-51 will take you to Cumberland where you'll pick up US-40, a scenic and historic route that will cut across southwestern Pennsylvania, the northern panhandle of West Virginia, and bring you into Ohio. US-250 is your next target and you'll pick it up just west of Wheeling and follow it northwest to New Philadelphia and Ohio Amish Country, then finish up your drive on I-77 to Cleveland. Besides those listed above, other sites of interest on or near this route include Harpers Ferry, Fort Necessity National Battlefield, and (although probably not an interest of your daughter) the NFL Hall of Fame.
You should probably spend two days on each of your legs, to and from Cleveland, to make this a true RoadTrip. That would leave you one full day in Cleveland where, besides the Rock and Roll Hall of Fame, you and your daughter might also want to consider the Natural History Museum or Cuyahoga Valley National Park.
AZBuck