Charleston, SC, with wife and 1 year old

We're looking at going in mid-June, from Louisiana. We could go to Destin or Gulf Shores like usual, but while we have the opportunity, we want to go somewhere far and different.

Any recommendations on what part of the city to go, stay away from, preferred pub type bars, attractions like zoo, etc.? We'll do probably a couple evenings at the beach, some tourism one day, some shopping or downtown sightseeing another.

We're probably going to rent a car, FWIW, so ubering isn't an obstacle.