1 hour from Asheville is Graveyard Fields. The best time to visit it is in late August, when it’s dripping with wild blueberries; but even now it’s a gorgeous place to hike. Go to the lower falls for some great views, and if you walk along the tiny river’s bank, you’ll come to a couple of great swimming holes. Go further for fewer people
The Nature Center is a small zoo of local animals, but they’re kept under very good conditions, and all the animals are rescues. We go there all the time. Definitely worth a visit if you like that sort of thing.
Linville Caverns are about an hour away, and they’re about fifteen minutes from Linville Gorge, with its own waterfalls.
You can go to Zen Tubing to rent inner tubes. Float down the river and end up at The Bywater, a bar with a huge lawn next to the river. Check on memberships first–NC has weird liquor laws that require you to get a membership at bars that don’t serve food.
Want something artsy and slightly goth and slightly retro? Crow and Quill is a cool but hard-to-find bar on Broadway.
There are a ton of excellent restaurants and brewpubs in this town. What kind of food would you like?
The NC Arboretum isn’t too far away–maybe 15 minutes from downtown–and is a great botanical garden with lots of hiking trails, a bonsai exhibit, etc. On your way back you can stop at Papas and Beer, a pretty good-to-great Mexicali joint.
Asheville Treetops Adventures and Navitat both offer up-in-the-trees ropes courses. I’ve been to the former (with 40 third graders), and it was a lot of fun, and I’ve heard good things about the latter.
Malaprop’s is a pretty cozy little independent bookstore downtown, has won awards for its quality.
Just walking around downtown can be fun. There are always buskers looking for your tourist dollars; flip them some!