The Charleston Waterfront Park is a pretty new park as far as Charleston goes, and is situated right downtown. You can see the Yorktown, Castle Pinkney, and the new (and old) Cooper River Bridges.
There is the huge pineapple fountain (pineapples are a symbol of hospitality) and the main fountain where there are always kids playing. It's okay--let them play there.
On the 'boardwalk" there are swings and a large sitting area where you can sit or fish or just look around. The park is very large and has plenty of room for you to find a peaceful spot away from everywhere else.
Beware--this is often a stop for school field-trips and pre-and-post Prom, so depending on when you go, you may be the only person there, or the only person over 18 and not wearing formal.....