Definitely get to NoDa in Charlotte. In Raleigh, I would hit up at least one bottle shop. The Beer Dispensary in Apex is wonderful. They're small so they're not riddled with beers like Sam Adams and such. Triangle Wine is second, they do have SA but their local section is not any more impressive than The Beer Dispensary. Both places will serve you well but I'm a fan of the Beer Dispensary.
I would hit up Raleigh Brewing Company, it's just a hangout of mine and my LHBS to boot. I would check out Trophy Brewing for sure, they're doing wonderful things. Crank Arm is pretty great and around the corner is Tasty Beverage. Basically in Raleigh alone you can't go wrong.
For a true English brewery, go to Fortnight Brewing in Cary. It is but 10 min from Raleigh. You can check out Big Boss and Lonerider but they bottle everything so you can try them just by picking up some bottles. Raleigh Brewing only bottles some stuff. If you want to venture past Raleigh into Smithfield, you can hit up Double Barley. Be prepared to maybe get a flight only. They're pretty high gravity beers. Or have a DD.
Natty Green's is a good suggestion but I prefer places with other beers on tap. I just went to T.Mac in Cary which has a ton of local beer on tap (and not local). Tyler's Taproom in Durham (my favorite), Raleigh, or Apex will all serve you very well. Both have full menus as well.
There is Clouds Brewing around the corner from Natty's but to my knowledge they're not serving their own beer. I went there opening night and it was a **** show. I figure I'd return when they begin to brew their own beer. They do have a nice selection of beer though. Food seemed a bit pricey.