Raleigh or Durham? They're about 45+min away from each other.
Raleigh has good museums that are free (NC Art, Natural Sciences, History), lots of options for great food & drink (local breweries, state farmer's market, great parks (state, city, & county). There's a lot of food you can't get anywhere else (e.g. Fiction Kitchen, a mostly vegan restaurant, down the street from a super-popular local BBQ place called The Pit; Escazú Chocolates; Triangle Raw Foods pop-up restaurants; a traveling food market (Lomo); There are great pizza options here too). The NC State Univ campus can be pretty to visit. First Fridays are amazing in Downtown Raleigh... art scene here rocks.
Durham can be more expensive on the museum side, but the Life Sciences Museum is pretty awesome. Food will probably cost about the same (food trucks are amazing here). Go to a Durham Bulls game, visit the area around it, see a show at DPAC, and drink a beer from Fullsteam, check out Duke's campus (nice hike around the forest). almost 5 years ago