North Carolina is an up-and-coming area for winemaking (http://www.ncwine.org/). The Yadkin Valley American Viticultural Area is a relatively new wine-growing region in the northwestern part of the state. One particular specialty of the state is wine made from Scuppernong grapes, a fragrant variety of Muscadine, which gives it a remarkable flavor.
Not a wine, but named as if it were owing to its burgundy color, is local cherry-flavored soft drink Cheerwine (http://www.cheerwine.com/). It's been a North Carolina favorite since 1917, originating in the town of Salisbury. Until recently, it was difficult to find outside of the area, but its popularity has caught on and it's beginning to expand throughout the US.
Another drink native to North Carolina is Pepsi Cola. It was made by a pharmacist in New Bern named Caleb Bradham who sold it in his pharmacy in the early 1890's. It was first called Brad's Drink but was changed to Pepsi in 1903.
And, of course, there's always the ubiquitous Southern sweet iced tea. As in practically all of the South, sweet tea is the beverage of choice for a lot of people; the stronger and sweeter, the better. "Iced" is always assumed ask for "hot tea" if you want it steaming and "sweet" is the default, although people still tend to specify "sweet tea" when ordering. Most places do offer "unsweet" tea, but remember to ask for it if you want it.
The alcohol laws of North Carolina prohibit the sale of alcohol after 2AM Monday through Saturday, and from 2AM until noon on Sundays. Beer and wine are available for purchase at most markets, grocery stores and gas stations but keep in mind liquors are only sold at state-run ABC Alcoholic Beverage Commission stores, rather than at a traditional liquor store.