Dan's suggestion of the Blue Ridge Parkway is viable, but it's going to be tough -- you'd start by going over the highest point of the BRP, a bit over 6,000'. A hard way to get started!
Do a search for Mark Boyd's tours, as he's done a fair number starting out of Asheville. I'd get a brother to drive to Asheville or preferably Mars Hill if you're going over 23 or 19. The direct route from Waynesville to Hot Springs is way curvy and hilly, from what I hear -- 5 hours drive for 30 miles!? I-40 west is a no-go unless you hitch a ride to Tennessee; no shoulders, heavy traffic, and BTW it's illegal on a bike (for good reason).
19E can be a bit busy for my taste, depending on day of week and time of day, but it is beautiful. Avoid NC 226 unless you're a glutton for punishment.
(Disclaimer: I haven't gone over 23/19W since they started interstate 26, though I'd expect low / local traffic on 23.) If you take 23 or 19W into Tennessee, TN 91/133 through Shady Valley is quite pleasant after you get out of the Elizabethton suburbs, and takes you straight into Damascus.