We decided to go to the Charleston Museum after visiting the Heward-Washington House and got a combined ticket. You can use the combined ticket over several days which made it convenient. The museum itself has the honor of being the first in the United States, albeit in a different building than where it is now. Most of the exhibits are on the second floor. One thing to keep in mind is that there is no restaurant or café so make sure to plan accordingly.
The museum has several different collections from the early ages until today. At some point the museum decided to only collect artifacts related to South Carolina, but still maintain and display those artifacts in its collection from its inception. The layout can be a little strange as it skips generations in the same room, one going from the Revolutionary War to WWII and back to the Revolutionary War. They even have a section dedicated to a naturalist who chronicled numerous birds and animals. There is a large selection of stuffed animals from around the United States and elsewhere.
They also do a good job of highlighting Charleston's history with the production of rice and the issue of slavery. They do not glance over slavery but show how it factored into history and the problems it caused.
If you are looking for something to do and are interested in history, the Charleston Museum is definitely worth considering.