Ashville NC and Greeneville SC
I live in Greenville and spend part of just about every weekend in Asheville. My first advice is to try to get more food in A'ville, less in G'ville.
JKidd is right, there a TONS of Asheville threads, give a search and then you can narrow down your choices. For Greenville, there's a lot less conversation because the chocies for deliciousness are fewer. That said, I would make sure to sample the Neue Southern Food Truck (you can eat your treasures at The Community Tap) and the Owl (again, search to find out what you're getting into.
Best ethnic - probably Mekong, Pita House also good.
Best Southern - eh...I haven't been able to get in Tupelo Honey yet, but if it's like the Asheville outpost, that might be it. For "fine dining"...nothing excites me much, but you could consider American Grocery, Lazy Goat, Northampton Wine Cafe.