For Charleston:
The ship will port at the Union Pier and Cruise Terminal, which is right downtown, within a very short walk to the historic district.
The Market is at the head of Market Street and E. Bay Street. You will find local vendors selling art, crafts, sweet grass baskets, etc. it has historic value, as the land was deeded to the city for use as a city market in perpetuity.
Horse Carriage Tours can be found outside the Market along N. Marke Street.. A great way to see the historic district.
A bit of trivia: the cobblestone streets downtown were paved with the ballast stones that were offloaded from ships porting in Charleston.
The US Customs House is an architectural masterpiece. There is the Charleston Aquarium, which is good. You can also take tours to Fort Sumter from the Aquarium.
Not in walking distance at the other end of Bay Street, the name changes to The Battery where the stately mansions of Charleston face the harbor and the sea wall. Past the Battery is White Point Gardens. This is where the Pirate Steed Bonnet and his pirate crew were hung, as well as over 200 others over time. This is where the first shots of the Civil War were fired from
There is plenty more to do and see in Charleston as it is filled with restaurants, historic houses, etc.