SC now has reciprocity with GA so as long as you have a valid GA CWP, you're good to go. Just pay attention to any businesses/buildings that may have valid "no carry signs" posted.
The aquarium, while not nearly as big as Atlanta or Baltimore, is a VERY nice aquarium. As also suggested, Patriot's Point has some cool sights. There are plenty of restaurants (all types of cuisine and varying levels of expense) in Charleston. If you're a PSA fan, they have a store in Mt Pleasant and their newest (and biggest) store is in Summerville and that one has an indoor range. Both are not too far from downtown Charleston. If you're a music fan, check the schedule at the Music Farm and see who's playing while you're in town. You can do harbor cruise/dinner cruises. Of course you have the Market downtown along with a plethora of other shopping destinations.