Live in the Raleigh area and go to the mountains as much as we can. Asheville isn't really our scene (Biltmore is cool to see once I guess but pricey and honestly rather do mountain stuff anyways) so can't offer much here. Blue Ridge Parkway is a must and amazing. We stay in Hendersonville at times (30 minutes SW of Asheville) and head to Dupont State Forest. There are 3 awesome waterfalls with relatively easy and short hikes connecting them all....Hooker, Triple Falls, and High Falls (there are others but another 5 miles of hiking). Part of Last of the Mohicans and Hunger Games filmed at these falls. Heck, Triple Falls is one of our favorite places period....had a morning picnic by ourselves one Monday morning there.
Brevard is awesome with lot's of waterfalls as well, and about another 30 minutes from Dupont. Pisgah National Forest is there and you can drive in and find Looking Glass Falls right on the side of the road and other awesome hikes and waterfalls everywhere around Brevard.
The Blue Ridge Parkway meanders somewhat around all these locations (may have to drive a bit to get on) and again is so awesome.
First pic is just a crappy cell phone pic on the Blue Ridge Parkway near Brevard (40 minutes) that doesn't do the scenery justice at all. 2nd pic is Triple Falls. Each fall is several stories high for scale reference and you can hike all around it.