Max Patch -- I think it's a state park -- is about 30 minutes from Asheville by car, and worth a visit. It's a mountaintop that was turned into pasture, and is now simply grass covered. Terrific vistas, and spatially interesting. Easy access by car; 1/4 mi (?) uphill walk. (If my weekend back-ups finish, I'll post a snapshot.)
The mountains around Asheville are extremely photogenic. +1 on the BRP, but it's hard to go wrong. Get a GPS for the car, and try every road that climbs. If you do drive the BRP from Asheville, I suggest northeast rather than southwest, unless you are planning a full day out there.
West Asheville is a small neighborhood that has an interesting urban, vaguely alternative, street scene.
There are some interesting industrial/railroad/river views along the French Broad River. Access (and parking) at either Carrier Park or French Broad River Park.
Have a meal at 12 Bones. Casual and delicious.