Gotcha, well, Raleigh isn't my specialty so much (although I will plug The Angus Barn, but that's just because I've been telling myself I need to try it for about a year now and haven't done it
), but I have a pretty good handle on the other two. It really depends on what you're looking for.
Durham -- Tyler's is a solid suggestion and a personal favorite of mine because their beer selection is so amazing (and their staff is really knowledgeable and can point you to some gems) and their food, while typical bar fare, is consistently solid. In that same vein is Bull McCabe's Irish Pub, though it's significantly smaller. BBQ is an experience many people want to take part in when they're in NC -- because of that, I'd suggest Hog Heaven Bar-B-Q for that traditional Carolina style, though I'd provide the disclaimer that I generally don't care for vinegar-based sauce so this recommendation stems from friends enjoying the place. Durham has a strong Hispanic population meaning that there are an abundance of outstanding, cheap Mexican joints to be found throughout the city. Two favorites of mine are El Cuscatleco and Los Comales. You'll most likely be one of the few native English speakers in the place, which is a pretty solid endorsement in my estimation. Those two places make great lunch stops, but my personal favorite lunch is Bull City Burger and Brewery. It's exactly what it sounds like, and they do it really well. Everything there is organic and done in house (they grind their own beef, even), and they always have a good selection of their own brews on tap to take the edge off. Great spot. One last local establishment that I'll plug is Dame's Chicken and Waffles, it's always a packed spot and for good reason. They do the classic urban dish with a number of innovative twists, and I've never had a bad meal there.
With those low-brow treats out of the way, it's worth noting that Durham has a number of more upscale options as well. Two options that readily come to mind are Four Square and Nana's. Both places have small but rotating menus that feature the work of really outstanding chefs that clearly have a passion for what they do. I've been to both places on multiple occasions and have always found my meals to be excellent and the wait staff incredibly helpful. Blu Seafood is another option that also makes a nice afternoon stop to have a few cocktails with their half price apps. Parizade and Vin Rouge also generally get a lot of love, but I don't care for them as much. Fairview Dining Room @ the Washington Duke Inn is another popular option, but I can't give it a full endorsement as I've not been there personally. Watts Grocery is probably a step-down in terms of "fine" dining, but that's no knock on it as an establishment. Solid food, although you should not order the jambalaya if it's on the menu. They clearly don't know what jambalaya is supposed to be, plus it had tomatoes. I was not happy
Other Durham highlights include Sam's Quik Stop for all of your beer purchasing needs -- amazing selection and you can probably find things there that aren't available wherever you're from. If you can't tell, Durham is a huge beer city, which is to be expected given its proximity to so many local breweries. Another store, Wine Authorities, specializes in stocking whatever is the wine equivalent to beer's craft brewing; a lot of bottles from small vineyards that are super "green." Worth stopping by if you want to pick up something that's probably different from your run-of-the-mill selections.
That's all I can think of off the top of my head for Durham, I'll be back later to give you some Chapel Hill options as well. I hope you enjoy your stay.
ETA: I don't know where you're staying, but if you're in the Southpoint area there are a number of restaurants in The Streets at Southpoint that are okay, I guess. I didn't recommend it initially because I assumed that if you were in the area you'd probably want to try some things that are unique to the area. If you'd rather stay in your "comfort zone" a little more there are a number of chains, I believe there's a Bonefish, P.F. Chang's, Maggiano's, Firebirds Wood Fired Grill, things like that. No judgment if that's your thing.
This post was edited on 2/24 at 2:43 pm