There is the Folk Art Center on the BRP north of Asheville. Beautiful work displayed there. Then a little further north is Mt. Mitchell, the highest mountain east of the Mississippi. It has a nice restaurant there too if you want some lunch. Beautiful views. Maggie Valley and Cherokee are interesting enough too. Wheels Through Time is a worth while stop if you like old motorcycles and cars. Lots of antique stores too. Museum of the Cherokee Indian is in Cherokee that is also a good stop. You might head a bit south and go into Lake Lure area. That area has Chimney Rock and nice little town and stores. Hendersonville is an old town with lots of antique stores
If you want a really nice dinner with some good beer there is the Sierra Nevada Brewery out by the Asheville airport. There is a lot to do in that area but I will say that if you have never been to the Biltmore House then you really should do it. Nothing else like it. Great food on the grounds of all types and plan on all day to see things. Take one of the add on tours like the roof tour or a behind the scenes tour they are very interesting. Bring comfortable shoes though, lots of walking.