Chimney Rock is about 25-30 miles east of Asheville (depending on route). Google it. Cool place to visit, and great view from the top. SR 74a west back to Asheville/Blue ridge is a nice ride ride too.
If you go there through SR 64, you'll go through apple orchard country.
This was a great low budget excursion.
Almost anywhere is a good ride chimney rock lake lure North into Marion north Cove. If your that close I would ride on up to mount mitchell. Then north on brp to Crabtree falls then. On up into elk park to elk river falls it's all good riding i see it every day.
You can grind a set of tires off down there & never get bored.
Try eating at some of the little mom & pop BBQ joints you see out in the boonies. UMM GOOD!
Lake Lure boat ride is nice..see if they are running.
Chimney Rock had a discount for AAA card holders.
The Biltmore Estate is nice...they have online pricing at a discount...it is a full day if you walk the grounds.
The rest has been mentioned by others.
Stop in one of the welcome centers and pick up the brochures. We go for a week each fall and never run out of things to see and do. Did the waterfalls last year, great riding and great sights.