I would suggest you wander around downtown Charleston in the Historic District. There is the City Market and French Quarter area. Lots of museum's in that area as well. James Island has some nice beaches and restaurants. Isle of Palms has nice beaches too.
Page's Okra Grill in Mount Pleasant (by Patriots Point) is DELICIOUS!
D'Allesssandro's Pizza is good.
Mondo's Italian is nearby James Island County Park and is delicious Italian food.
Take a walk or bike ride down East Battery to see some GORGEOUS Antebellum style homes.
A horse carriage tour of the city looks fun, we haven't done it yet though.