Make sure you get reservations as the nice places tend to fill up during Market.
You can't go wrong with the Proximity Bistro, Green Valley Grille, or Liberty Oak for fine dining. There is a place called Bin 33 that is the new thing; their menu is unorthodox but it's pretty good. For really high end dining, there is a place called Marisol in a little nondescript shopping center on High Point Road near Sedgefield Country Club that is expensive but FANTASTIC.
For less expensive but still good food with a more casual atmosphere (and good beer) there is Natty Greene's, Grey's Tavern, and M'Couls, all on South Elm downtown. Also you could go to the corner of Walker and Elam and check out Fishbones for some excellent fish tacos or Filling Station for great sandwiches and burgers. Also at that corner is Sticks and Stones which is owned by the same people as Fishbones and makes really good gourmet pizza.