We spent spring break in the Charleston area 2 years ago when my son was 6 and my daughter was 8. We went to the aquarium. I remember thinking that the aquarium was nice, but we have definitely been to better ones. My kids loved it though, so I know they would recommend it.
We also went to Fort Sumter. My husband, daughter and I really liked it. (I'm pretty sure that was the day my son was being a pain, so his opinion doesn't count.) There is also a battleship that you can tour at the same dock as the boats to Fort Sumter. My husband really wanted to do that but we ran out of time.
We went on a walking pirate tour around Charleston. (http://charlestonpiratetour.com/default.aspx) We all like learning about pirates, so it was fun for us. (Plus we thought the kids were still a bit young for a ghost tour.) The tour guide dresses in full pirate gear and carries a real parrot as long as the parrot behaves. A lot of people stop and stare as the tour group goes by. The tour ended up being longer (in both time and distance) but the guide was very informative and we got to walk past some beautiful houses.
We also went someplace that I can't think of the name of right now and I can't find my guidebook. It had a small zoo of animals native to South Carolina. It had a replica of a trading ship. And it has a replica of a small village from way back. And I believe there was a small museum at the beginning of it. Both kids enjoyed the zoo and the replica of the ship. My daughter enjoyed the museum too.
We were there the first week of April and it was warm enough to go swimming in the pool a few days. The kids also played in the ocean a bit but really didn't go past their knees. There were people swimming though.
We also went into Charleston and walked through the City Market on Market St. There are all kinds of shops along there and you will see people making and selling the sweet grass baskets. Some of the baskets are pretty amazing. If you decide to buy one, look around because the prices vary quite a bit.
We are going back this year and I am planning on going to a plantation. (That was one thing we didn't get in last time that I really wanted to do.) The owner of the house I'm renting said there is one that she and her kids like a lot but can't remember if it's the Magnolia or Middleton plantation. She's supposed to get back to me with which one it is.
Last time we stayed in the Wild Dunes resort area on the Isle of Palms. This time we are staying on the Isle of Palms again but we are staying outside of the Wild Dunes area. It's a beautiful area and a short drive into Charleston. And we found a way into Charleston that cut across the islands so we were able to avoid the heavy traffic area.