Word of warning though - if you're social justice/equality minded, Chick-fil-A may be a place to avoid. Not sure if it was in the news much, if it all, in Canada, but the owners of the chain are evangelical Christians who donate to anti-marriage equality and anti-gay gruops pretty regularly.
Bojangles, that DaveG mentioned, is another fast food chain. Not exactly a place to get a "real" Southern breakfast, but not bad. For the "real Southern breakfast experience" I'd recommend going to the State Farmers Market Restaurant or Big Ed's City Market in Raleigh or Mama Dip's in Chapel Hill. Cary is technically a separate town from Raleigh but for all intents and purposes, it's a suburb - the line as to where Cary begins and Raleigh ends can be fairly murky at times.
As for what to try at a barbeque place, if you go to The Pit as has been recommended I usually go for their Carolina Classic Combo (chopped pork and a piece of fried chicken. I usually eat my pork and take my chicken to go to serve as a meal later) which comes with your choice of two sides. All of their sides are great - the baked beans are made with a little moonshine and I'm also a fan of their mac-n-cheese. At the table are two bottles of sauce - the darker, redder sauce is Western North Carolina style sauce which is tomato based. The lighter, more liquid sauce is Eastern North Carolina style sauce which is vinegar based. That's the most authentic to this region, but I encourage you to try both so that you can pick a side in the great barbeque debate. If you get dessert, there's only one truly authentic choice - banana pudding.
Lastly, if you're a beer fan at all, visit one of our fine local breweries.