Since you are traveling from NC, recommend going past the Badlands, through the Blackhills, across the Bighorns on Highway 16. Finally stop at Cody for the night before you enter the park to relax, restock, and get a morning start into the park. My recommendation then would be to just take the short route into the East entrance and stay at Bridge Bay. Canyon Village is a bit more centralized but I think Bridge Bay is much prettier with a bit more open so easier to see the wildlife that goes through the park.
Figure at least 5 nights in the park or it isn't worth going. I've been twice at five nights each and will be going back again within two years. Next trip I'm planning on two weeks in the park and hope that will satisfy me for another year.
While all the routes are very scenic, the most scenic would probably be to take the Chief Joseph Highway out of Cody into the north east entrance. But the scenery is really inside the park so only take this route if you have plenty of time. If you go this route, consider staying just outside the park for one night which would give you a chance to unhook and go see the views from the top of Beartooth Highway.