I'll speak some for the Triangle area (Raleigh-Durham-Chapel Hill).
I've been in and around the Triangle area for most of my life, and can say that it's a great place to live. Summer is tough, though, with the humidity and all.
Aside from Wilmington and New Bern, the coast is pretty much all tourism. The eastern part of the state is fairly poor, so it's not until you get to the Piedmont that things start picking up, educationally speaking.
Durham, in the Triangle, is a pretty interesting town with education, startups, tech companies, culture (did you know there's a Durham style of blues?), food, museums, a living downtown area, etc. It's worth a visit.
Carrboro is has the greatest population density, and is the most Mustachian of the towns. Really interesting place. I get the feeling Bakari would like it there. :) Lots of people walk or bicycle everywhere, and they have an amazing farmer's market.
Hillsborough is near where I live and has a lot of history, as it was temporarily the state capital during the Revolution. It also has an interesting downtown area.
Raleigh has good food, the NC symphony, and a bunch of other things you'd expect from a rich, somewhat culturally conservative city. Museums are good, downtown is fairly well functioning, but it has a tendency toward sprawl. It definitely has liveable areas.
Chapel Hill is like Raleigh, but full of rich Democrats who drive Land Rovers and Minis and shop at Whole Foods instead of rich Republicans driving Ford Excursions and Lexuses (Lexi?) and shopping at Harris Teeter. That's something of a cheap shot, but it's how I think of the place. :)
I live in a small town west of the Triangle and spend most of my time here. Very quiet, very provincial in many ways, despite being a bedroom community. I find it very bikeable, but I work from home, so a commute isn't an issue for me.
As for things you shouldn't miss, it depends on what you like. Mebane, Hillsborough, and Carrboro are all worth wandering around a bit, but they're places to live, not visit, usually. We have a number of nice state parks, like Pilot Mountain (near Winston Salem, but a ways from I-40), Hanging Rock (also out of the way), Carolina Beach (near Wilmington) and a Whole Lot More! Nice museums in downtown Raleigh.
The trouble is, whenever someone comes from a long way off and asks what there is to do, I have a hard time answering. It's more of a nice place to live than a nice destination.
I'm happy to answer specific questions. I love living here. :)
PS - Asheville is very nice. You'll enjoy it.