Since Governor Sanford is trying to refuse bailout money for the state, I figure I'll help out the tourism industry as best as I can.
Riverbanks Zoo - In Columbia, this is a truly great zoo. It can be a little crowded, but is large enough that the crowds don't usually feel pressing (unless you're at a feeding or event) Check out the gardens if you want to steer clear of the beasties, but be sure to check out the penguin feeding and the special area where you can feed tropical birds little cups of nectar.
Skip: Ripley's Aquarium - Typically crowded, boring and overpriced.
Downtown Greenville - Greenville's downtown is beautiful and lots of fun. On any given weekend you can almost always catch some sort of concert, festival or show either on the street, in a park, or at the Peace Center or BiLo Center. Restaurants range from sandwich shops to high-end sushi. Make sure you check out Falls Park and look for a series of actual size statues of bronze mice hidden around the main drag.
Skip: Downtown Charleston - If you are really into Southern history or have a lot of money, ignore this. The place is pretty hopping but most of the good stuff can get pricey. If you go to see the market or shop at all make sure you go during the day.
Piccolo Spoleto - While downtown Charleston is kind of a bummer unless you have lots of money for shopping and food, not so during Spoleto. This huge music festival takes place in the early summer. Concerts of all types take place in venues across the city all day long, and performers and patrons flock from all over the nation.
The Cooper River Bridge Run - This is a marathon across the historic Cooper River Bridge in Charleston. Now it is a brand-spanking new bridge, and a pretty one too! Those are the older two versions in the background although I think they are gone now. Anyway, get the event t-shirt and you can use it as camoflage to masquerade as a SC native anywhere you go.
Skip: The actual "run" part of the run. Sign up as a "walker" unless you want to be trampled by Kenyans.
Enjoy the Food - Find at least one greasy diner (like The Beacon in Spartanburg, or Ye Olde Fashioned in Charleston), one thouroughly southern restaurant (like Lizard's Thicket) and at least half a dozen Krispy Kreme donuts. (If you see a red "hot and now" sign in the window it is time to pull over NOW.
Skip: The same old, same old. Promise me you won't head to Olive Garden when you're here.
Go to a Festival - Almost every small town will have one in the spring and many hold them yearround, it's just a matter of knowing when they are. Sometimes they will be specific - like the St. George Grits Festival - and sometimes just named after a plant or something. Anyway, these are usually an entertaining afternoon.
Magnolia Plantation - One of my favorite places in the world. Find this on the outskirts of Charleston outside of West Ashley. The gardens are beautiful year round, but when the azelias bloom in the spring is by far the best time to visit. Don't think cultivated flowerbeds, think towering bushes covered in blossoms, wooden bridges in lush forests, and generally a Secret Garden beautifully overgrown feel. There is also a series of walkways through the swamp (never get your feet dirty), a hedge maze, petting zoo and I assume some sort of museum.
Skip: Drayton Hall Plantation - It is literally down the road from Magnolia, but is not as fun or spectacular.
The Mountains - Check out Ceasars head, near Brevard or one of tons of other mountain-area parks and activities.
Go Strom Hopping - Everything here is named after Strom Thurmond. Check out Lake Thurmond, buildings on the campuses of nearly every major public university, statues of him at the capitol and his hometown (also the hometown of Strom Thurmond high school). Actually... this is not that fun. The lake is nice though.
Bull Island - A coastal island which is nearly pristine. You'll have to take a ferry to get there. This is one of the few places in the world where you'll have a chance to see a red wolf. This is a little more of an expedition than a tourist sight though, so don't bite off more than you can chew if you can't handle the bugs or the heat.
The Beach - We have some amazing beaches!! The tricky part? Avoid Myrtle Beach. Seriously! It's a fun city to hang out in, but the beach is so polluted crowded that it is no fun for actually be on. Check out Surfside, Edisto, Isle of Palms, Folly, and Ocean Isle (actually in NC but whatever). There are certainly others but ask someone who knows what they are like before you head on down.