Take a stroll along the water at Waterfront Park. You can also see the iconic Rainbow Row a few streets away.
Walk around and window shop the various shops on King St.
One of the beaches near Mt Pleasant is Sullivan's Island. It's a small/private beach town with a quaint Main Street. Drop by "Home Team" for some of Charleston's favorite BBQ.
Another beach town close to Mt. Pleasant, IOP, is a popular tourist attraction with easy access to parking, shops, and restaurants.
If you'd like to grab a drink by the water, head over to Shem Creek. It's a lively stretch of bars and restaurants along an inlet. You might be lucky enough to see dolphins!