Still not quite clear if the idea is to fly in, or to rent once here. If you're flying in, and going to Chapel Hill, you may as well fly directly to Chapel Hill (IGX). They are constantly on the verge of shutting down in the political/university world, but they're definitely fully operational at the moment, and probably will be for a while.
I've also flown into RDU from afar, and it's very easy, as really it's a quiet field with very accommodating controllers. I used Atlantic in the past, but didn't do much comparison. It wasn't terribly expensive compared to other "big" airports.
If the idea is to rent once you get here, and you don't want to drive 45 minutes to do it, your options are more limited. I can recommend FlightGest at RDU. They'll require a check-out of course, but are very friendly. And they're on the expensive side (I forget the exact numbers, but figure $150/hour for a G1000 172), but for a one-off flight you might not care. They're a reasonably-sized flight school, with all of the advantages and disadvantages that come with that. There's also Blue Line Aviation at RDU. Don't know too much about them.
Further afield, there's Empire Aviation up at Oxford/Henderson (HNZ), Burlington Aviation and the (single-plane) Elon Aviation at Burlington Alamance (BUY), and a couple of options I know nothing about down at Sanford (TTA). But honestly, if you're trying to impress your girlfriend, I'd just eat the extra cost and go with FlightGest where I know the operation will be professional (front desk and everything) and the planes are presentable.
In terms of where to go flying, if you or your girlfriend have any connection to Duke or UNC, the RDU controllers are happy to let you do air tours of the area without any trouble (the C is very lazy). (They'll be happy to let you do air tours if you don't have any connection too, it just might not be interesting for you.) You could also fly down to Carthage (5NC3) where there's a decent BBQ place right on the field, with outdoor seating looking out on the runway. If you have a little more time and want to make a half-day trip, get out to the outer banks (Ocracoke, W95 for beach/town fun; Kitty Hawk, FFA for airplane history) or Asheville (AVL) or something. And there's always the "just fly around" option!