If it's your kind of thing, check out Biltmore house. Pretty amazing.
http://www.biltmore.com/

 

Me on the far right:



As others have suggested, there's the Blue Ridge Pkwy, and the Cherohala. Both are very scenic and lots of fun on a bike. We rented a house in Franklin, NC last year (about 40 miles south east of Ashville) and went out everyday riding, always on a different route. Get a map, pick a road, and have at it. There are no bad roads down there. If you're into rafting, head out to the Nantahola River and spend the day on the water. We rode the bikes there, got wet, and rode back to Franklin.

I'm in the front of the raft, under water in this picture. lol

 

Not too far away in the town of Brevard is Oskar Blues Brewing (Dales Pale Ale) and Siera Nevada built an amazing new $100 million brewery near the Asheville airport. You can sign up for a tour on their website or just visit for beer and they have a restaurant also. Blue Ridge down and visit Chimneyrock for spectacular views.

 

Got the Roadglide in Hickory now we are in Asheville at the Grove Park Inn having a expensive brew. Then it's off to Marshall to meet a friend.

 

For sure check this out!!

 

Ah Ashville. A decade ago my wife drug me to Biltmore and we spent a week there one afternoon. If you have a significant other with you on your trip avoid that area like the plague.

 

I got a map from the HD place in Chimney Rock that had all the roads for great motorcycle rides, and they are a bunch.

 

Just completed a 1400 mile trek through East TN, West NC and N Georgia area...Beautiful area - hit maggie valley - wheels through time...Previous posters also had some great suggestions.