Asheville Brewing on Merrimon Ave
The Thirsty Monk on Patton Ave (My favorite bar ever (all belgian beers downstairs))
Jack of the Woods across the street from the Thristy Monk is OK
The Bier Garden is right across from Malprops bookstore, near the convention center.
Mellow Mushroom, Barleys Tap Room (great)
Look up the French Broad Brewery, small low key operation, but DANG GOOD beer.
Highlands Brewing is close
Pisgah is close
MAKE SURE to look up Brusin Ales, they're right off 240 Exit 5 downtown, the best bottle selection in the city.
If you want more suggestions, PM me, I go there many times a year.
They also have a great homebrew shop right across the street from Asheville Brewing.
Asheville also has a beer tour bus, but Ive never tried it.