Update:  thanks to all of you for the suggestions; I found that there were way too many things to do so had to prioritize.  Some of my favorites, though, were:
[ul][li]The State Museum in Columbia, SC, although the pirate display wasn’t really worth the extra money because I am older than seven.  It sort of felt like Disneyland had a 2 for 1 sale on Pirates of the Caribbean stuff, and Columbia went nuts.[/li][li]The Biltmore in Asheville.  Their website tells you, anxiously, that you really can’t do the estate in a day, and I mocked them for that, but really, a day isn’t nearly enough.  (I skipped Macon in favor of Asheville.)[/li][li]The Bellamy Mansion in Wilmington, NC, which wasn’t even in my guidebook.  I really hope that they can raise the money to renovate the slave cabin; only the ground floor is open so far.  But the audio tour was very good, and the house was amazing.[/li][li]Fort Sumter in Charleston, not just for its historical significance, but also for the brief, breezy harbor tour as well.[/li][li]Bonaventure Cemetery in Savannah, although I’d advise other visitors not to make the same mistake I did.  Remember where you parked the car![/ul][/li]
Overall, a great trip, and definitely a very different place from LA.  When I got back home, the trees here seemed anemic and puny for the first time.
That said, there are some strange things people in that part of the world do.  First, sweet tea.  Ordered it once accidentally, and I cannot believe anyone voluntarily drinks that stuff.  I do believe that every time I ordered UNSWEETENED tea after that, I saw a bit of a smile on the waiter’s face.  I also wondered what my accent sounded like to people I talked to.
Second, the Civil War.  There are some crazy folks out there, and I must admit that the tour of Fort Sumter was a breath of fresh air in more ways than one.  I had started to doubt myself about the Civil War (i.e., what the causes were), and it was such a relief to get on board the ship to the fort and hear the word “slavery.”  To hear many folk tell it, not only was slavery not a cause of the civil war, there were barely any slaves (but they were totally happy).  Very strange.  (One tour guide told us, “One of the worst things about the Civil War was that it destroyed this great lifestyle that everyone enjoyed.”  Um, not really everyone I think.)
Finally, you people are strange drivers.  No one really speeds there, except for a few people who go 20 or 30 above the speed limit; and then there’s that one guy toodling along at 20 below the limit, blocking traffic.  I also couldn’t tell what the pedestrian laws were; I saw pedestrians who stopped to let cars turn, and others who didn’t.
Savannah and Wilmington were my favorite places, and whether or not I ever get back there again, I’m glad I went.