Durham's downtown is the spot. American Tobacco Campus turned old factories into restaurants and breweries—Fullsteam, Durham Distillery, Bull City Burger. Motorco has shows, food trucks, and a patio where people hang out past one drink. The Durham Bulls play at a stadium that inspired Bull Durham, if that matters to you.
Raleigh's got museums—the North Carolina Museum of Art has this huge outdoor park, no admission fee. The Natural Sciences Museum is way better than expected. Warehouse District and Glenwood South have the bars, though weekends get pretty bro-heavy.
Chapel Hill is UNC's campus, Franklin Street with bars and cheap food, and that old college- town feel before everything turned into luxury buildings.
BBQ matters here. The Pit in Raleigh does whole hog. Clyde Cooper's has been there since 1938. In Durham, hit Picnic or Backyard BBQ Pit. People argue about which style is correct—just eat and quit overthinking.
Spring (March to May) and fall (September to November) are your best bets. The weather's decent, everything's running, and college sports happen if that's your thing. Summer's hot and gross. Winter's short and fine, and you'll get the cheapest flights to Raleigh-Durham then, mostly in January and February.
Raleigh-Durham International sits between both cities, maybe 20 minutes from either downtown. Rideshares run $25-35, depending on where you're going. There's no real public transit to the airport, so you're paying for the ride. For cheap flights to Raleigh-Durham, check numerous budget carriers—they've been piling on routes as the area blows up. The Triangle isn't flashy. It just works way better than it should.