Sullivan's Island is about a 15 minute drive from downtown Charleston, and about 45 minutes from Summerville. Sullivan's Island has pubic beach access.
Park along the street around White Point Garden. From here you can explore the park and walk along the seawall. If you drive down Murray Boulevard you will have plenty of houses to "ooh" and "ahh" over. The homes along South Battery and East Battery are some of the most iconic homes in Downtown Charleston, many of which pre-date the attack on Fort Sumter in 1861.
The Charleston Place is a good central location when exploring downtown Charleston. They have a parking garage with easy access to both King and Meeting Streets. The lobby of the hotel is open to the public. It is a great place to cool off and they have public restrooms.