Beautiful place. That is my neck of the woods. Shopping in the market is a must. Get a crepe at Queen Street Grocery. If oysters are in season when you are there, the 167 Raw Oyster Bar has a wide selection. Church and Union has great brunch. Walk the Battery.
If you want a fancy meal, Tempest, 82 Queen, and Fig are my favorites.
If you've got a day and want to get a different, more chill vibe, Folly Beach is great. The Lost Dog Cafe there is by far the best breakfast food. The blueberry muffins are excellent. As well as the biscuits and gravy (add Sriracha), the breakfast burrito (big enough for 2), and the juevos rancheros. Bert's Grocery is the local lunch spot and a great place to pick up beers. If you are going to drink on the beach, just don't do it near the pier and don't make it obvious. It's technically illegal to drink on all beaches in the county, but unless you are being a problem, they don't check.
Feel free to PM if you have any specific questions. It's one of the coolest cities in the country by far.