Drive directly to Chattanooga by interstate, you are wasting time until you get there. Then go ahead to the Tail of the Dragon go around the bottom of the park to Sylva, NC continue over to Asheville and drink beer at the local breweries. Take the Blue Ridge to Spruce Pine, take 19E to Cranberry then 194 to Linnville. From Linnville take US221, great road good condition and no trucks, to Blowing Rock stop and have lunch. Now Shulls Mill road to Hwy 105 go left to 184 right to Banner Elk right on 194. If you stay on 194 its a nice two lane for a few miles then goes to a four lane through Boone, you will turn left to stay on 194 and it continues as a two lane for many miles. When you get back to US 221 turn left and it will take you up into Virginia and back to the interstate. You will be back at the Blue Ridge parkway if you want to continue to Skyline drive. Many other good roads in this area, but this will get you through NC and VA if you like.