There are so many rockclubs in Chapel Hill that there aren't many "normal" bars/restaurants that have live music. On some weekends you'll hear music coming out of West End Wine Bar or Tallula's. The two best beer selections in town are Tyler's and Milltown, but with the exception of Milltown brunch, and an irregular, semi-biennial parking-lot party at Tyler's, neither has live music.
I think Linda's on East Franklin has live music sometimes, but unless you're really into the whole "college town bar across from campus" thing, it may not be that much fun. There's also a weird newish place called Blend that tries to be a bistro, a coffee-bar, a danceclub and a rockclub, all at once.
Of the rockclubs, the best in terms of beer selection & general atmosphere is The Cave, and it has the added bonus of having 2 sets of live music a night, one at 7:30 and one at 10:00. Plus the front room (stage/bar) is nonsmoking, while the back (pool/jukebox) is smoking, so either way you're covered. They only have 4-5 taps, but they've got a good selection of bottled beer (and the taps are local microbrews or decent imports, not crap).
A lot of the interesting dinner options (Lantern, Elaine's, Tallula's) are in the same 2-block range as the clubs (Local 506, The Cave) (and heck, all of Chapel HIll/Carrboro is relatively tiny). Lately we've gotten a pack of those pedicabs, as well, so you can haul from one place to another in high style ;-)
Adding links for most/all of what has been mentioned so far, although the link-a-place thing is acting weird & keeps adding random restos I didn't ask for, so . . .
307 E Main St, Carrboro, NC 27510
1215 E Franklin St, Chapel Hill, NC 27514
Elaine's On Franklin
454 W. Franklin St, Chapel Hill, NC 27516
Italian Pizzeria III
508 W Franklin St, Chapel Hill, NC 27516
456 W Franklin St, Chapel Hill, NC 27516
101 E Weaver St, Carrboro, NC 27510
West End Wine Bar
450 W Franklin St, Chapel Hill, NC 27516
711 Rosemary St, Carrboro, NC
Merritt's Store & Grill
1009 S Columbia St, Chapel Hill, NC 27514
211Pittsboro St, Chapel Hill, NC 27516
452 1 2 W Franklin St, Chapel Hill, NC
Blend: Nightclub & Lounge
157 E Rosemary Street, Chapel Hill, NC
Acme Food & Beverage Co
110 E Main St, Carrboro, NC 27510
Chapel Hill has 1 Thai restaurant, and 1 Singaporean, which is sort of a cross between Thai, Chinese & Indian. Depending on where you're from and how you feel about authenticity in your Thai food, you'll either like Thai Palace just fine, or find it kinda blah. I really like Merlion, the Singaporean restaurant, with the caveat that the decor is a lot like that of a hotel lobby, and it often seems to be packed with elderly dudes in sport-coats.
Durham has 3 or 4 Thai restaurants, and they all seem to have suffered at various times from the spotty-service, inconsistent-food problem (which isn't the case at Thai Palace, which is nothing if not consistent, for better or worse). Of them, Thai Cafe seems to lately get the best/most consistent positive reviews.
Raleigh & Cary have a few Thai restos as well, and different people are partial to different ones. I've had nothing but good meals at Sawasdee, but I don't live in Raleigh, so I don't have a big enough sample set to be able to say anything definite.
If I were in Chapel Hill & really craving Thai, I'd probably recalibrate the taste-buds slightly & head for Merlion. In Durham, Thai Cafe. If I were closer to Raleigh, I'd still head to Sawasdee.
If I only had one night in Chapel Hill, and was looking for interesting Asian-inspired cuisine, I'd definitely be hitting Jujube or Lantern before considering any of those, though.
2501 University Dr, Durham, NC 27707
410 Market St, Chapel Hill, NC 27516
4201 University Dr Ste 112, Durham, NC 27707
Thai Lanna Restaurant
5410 Nc Highway 55 Ste F, Durham, NC 27713
2434 Wycliff Rd, Raleigh, NC 27607
Sawasdee Thai Restaurant
3601 Capital Blvd Ste 107, Raleigh, NC 27604
Simply Thai & Sushi
4731 NC Highway 55, Durham, NC
Thai Palace Restaurant
1206 Raleigh Rd, Chapel Hill, NC 27517
1319 Buck Jones Rd, Raleigh, NC 27606
If you want a quick breakfast, Weaver St in Carrboro (a ten to fifteen minute walk from downtown Chapel Hill) has amazing pastries and pretty good coffee. They also have hot food if you're looking for that. It gets crowded on the weekends so try to go early, as in before 10.
Merrits is open at 7am and a single layer BLT is the best breakfast option in town. The grill is closed on Sundays but every Saturday I hit it for a sandwich before I run my dogs.
Aside from their beer selection I believe Milltown to be mediocre these days (at best). It is pub fare just not what I consider to be tasty.
If you want something on the upscale side (with good food) I'd say Elaines or Lantern. Talullah's is also good, lower-key and fun atmosphere, Turkish food. These are all walking distance, which is always nice (more wine please!).
You didn't mention lunch, but if you have a chance, get yourself a sandwich at Sandwhich (or have I gotten confused on which is the restaurant name and which is the correct spelling?). The place is fantastic. Follow it up with a popsicle from Locopops (in the same building) and you'll have a wonderful lunch.
Chicken Biscuit at Sunrise as well as an immaculate cappucino at Driade (both on Franklin not far from downtown heading towards Durham you can walk it but it will take at least 15 minutes both ways) should be on your short list of morning options.
If you want to eat on Franklin, I would vote for either Lantern or Elaines. If you're prepared to venture out a bit, your options open up.
I'll second the Merrit Store for a BLT at lunch. Great sandwich.
Speaking of sandwiches, Sandwhich is right downtown and does a fabulous job. They're right behind Panang (a place that I don't endorse not only because the service can be horrible, but I think the food is average at best).
If you like beer, check out Milltown just towards Carrboro from downtown CH. Nice selection of brews and some tasty pub fare as well.
If downtown, Sutton's is the obvious choice for breakfast. Average food but great local color, esp. if you sit at the counter. Merritt's is good but that is a tricky trip w/o a car if you don't know the area.
For dinner? Any of the dozen CH threads on here should give you an idea of the choices. Any of those above would do you well, I'll add Crook's Corner and The Lantern to the list of obvious choices
Sutton's Drug Store
Chapel Hill, NC, Chapel Hill, NC
Chapel Hill, NC, Chapel Hill, NC
Penang has yo-yo service, but good food - SE Asian, right on west Franklin.
Dead Mule Club is a bar on West Franklin - Local flavor, no you don't have to be a member to check it out, tell them you're a visitor.
Spanky's is an old Chapel Hill institution, they now have antibiotic hormone free burgers that are quite good.
Il Palio at the bottom of the hill in the Siena Hotel is more expensive, but better than 411 West. Otoh, Panzanella is quite decent, and not quite so fancy.
The Carolina Inn downtown is worth a look, if you want to eat somewhere here that is older than dirt.
Red Lotus is good "neo" chinese, and they have an amazing bartender (the owner).
No one here recommends Weathervane at A Southern Season. It's a nice restaurant, with a bar area, and you can shop and wander the store as well (if you've never heard of A Southern Season, it's worth a visit). It's a wee bit chrome & fern as a restaurant, ok, but really the food is surprising.
Breakfast in this town is more difficult, depending on how early you want it. There isn't any place I'd "recommend" except Merritt's Store, which is on South Columbia Street down next to the 15-501/54/Fordham Blvd overpass. It is the only country store remaining in Chapel Hill, and they serve decent breakfast, but killer BLTs. No, they don' t have a sit down area, unless you call a bench outside sit down. Sutton's is the other alternative if you don't want to go the 6 blocks to Merritts.
It beats any chain to death, and the quality at Elmos & and the service/quality at Breadmans have gone down too much over the years for me to recommend anymore.
What is your price range? What atmosphere do you prefer?
For more upscale dining, I'd recommend Elaine's on Franklin, Acme in Carrboro, or 411 (though, that's more casual Italian).
For pizza, Pepper's or Italian Pizzeria III.
For burgers, Sutton's Drug Store.
For burritos, Carrburrito's.