Kiawah puts you in a paradise setting that you will need to leave in order to get to the locals. Also, remember that this is a tourist destination, so many of the people you will encounter will be in the same boat as you!
For sure, I would start with the Farmer's Market on Marion Square on Saturday morning. You can get breakfast or lunch while you are at it.
Here is a link that will keep you busy. Be sure to check the different areas (Music/Dining) in the search request for each day. Let me know if it does not work.
Events - Charleston City Paper - Charleston