One of my favorite places is Asheville, North Carolina. (Had family living in this area) It's a cool little mountain town with good restaurants, small family-run hotels and B&B's and a lively farmers' market on the weekends.
The must-see local attraction there is Biltmore House and Gardens, a French Chateau built by the Vanderbilt family in the 1800's as a summer residence. The house is spectacular, and the gardens even more so, having been created by landscape architect Frederick Law Olmsted Jr.
The mountains of North Carolina are spectacular, whatever area you visit.
Part of the Vanderbilt Estate lands were given to the state in lieu of taxes, and became the Pisgah National Forest. This is an excellent park with walking trails, waterfalls and campsites. Highly recommend. Mountain bikes are welcome on the trails, too. Great birdwatching.
Many movies have been filmed in the Pisgah National Forest, including one of my favorites, The Last of the Mohicans, with Daniel Day Lewis. You can walk the trail that they walked taking their prisoners up the mountain and stand under that waterfall where they hid. Amazing scenery, thick forests, no crowds, no restaurants or hotels to spoil the forests. The Cradle of Forestry is in this area.
The Blue Ridge Parkway runs north and south through this area. It's easy to jump on this and travel north into the mountains of Virginia or south into the mountains of Georgia.