The famous Blue Ridge Parkway goes from Asheville, North Carolina to the Shenandoah National Park in Virginia. It always follows the highest elevations of the Blue Ridge Mountains, part of the Appalachian Mountains, so you'll get nice views of the mountains along the way. Probably the best time to go there is late fall, when the air is dry and clear, so the view is best. In summer, the humidity is so high, that the mountains seem to vanish in blue smoke, that's also where the name comes from.
One of the highest mountains of the Appalachian chain is Mt. Mitchell in North Carolina, just northeast of Asheville. You can actually drive up to the top from where you have a fantastic 360 degree view.