My two favorite points are...
Route 7 at Weather Mountain. (aka: Snickers Gap)
or
Route 340 at the VA/WV border.
I like these spots because they both have decent parking areas to leave your vehicle for a few days.
A bit more advice here too. I would highly suggest you try a day hike starting at each of these locations to see how you like the parking areas and the starting points.
Also, do NOT use the shelters they have on the AT. Most shelters are located in close proximity to the highways. This means that the local riffraff like to use them for underage drinking and a place to use illegal narcotics. Take biker/hiker tents and camp in a nice safe spot off of the trail and away from the roadways.