>>2124577skyline drive-->blue ridge parkway-->smoky mountains national park
your basic route is all laid out for you. something like 600 miles of gorgeous parkway following the appalachian trail. if you want to explore a bit go down in the valley. if you want to really explore drive highway 60 through WV as well. this will take you through the New River Gorge (which was just given its own national park) and a unique part of the country. when you reach eastern tennessee there is a ton to see there. just add on whatever piques your interest. if you want to go further south you can cut across to nashville and go down the natchez trace a ways, another excellent parkway, less scenic perhaps but with more historical interest. there's also plenty to see in the harper's ferry area, western maryland, southern PA, though that's not really what we think of as appalachia anymore and you'll want to give DC as wide a berth as possible. I recommend early winter as there will be very few tourists and you can actually see more with no leaves on the trees. only potential complication is weather. usually not parkway-closing conditions but the mountaintops can be fogged in for days at a time. very atmospheric but not conventionally scenic.