and there is lot to do, depending on how much time you have, and what your priorities for your stay are.
If you want beach - Isle of Palms is gorgeous, and Folly Beach is nice too.
If you want unique, historical, etc:
No trip to Charleston is complete without visiting the market. The market is a combo indoor/outdoor shops and vendors. Everything from $3 touristy junk (which my 7 yo DD adores shopping for) to high end jewelry, housewares, clothing, etc.
If you have never done a carriage tour, I'd recommend it. If you've done one somewhere else, they are very similar. Fun, but not necessarily a must-do. Be warned, I've been on several, and no two narrations are ever exactly the same (poetic license is not unusual

)
A walk down the Battery is nice - it is the harbor front and the home of the famous "Rainbow Row" (the original old mansions all painted muted shades of peach, yellow, blue and green). Great for adults, but depending on the time of year, can be dull for kids and get them a bit hot and cranky.
Waterfront park is very nice. Basically a long pier with oversize porch swings, fountains for the kids to play in, and a shady park to enjoy an icecream cone. Near one end of the battery.
Depending on the age of your kids, we like CharlesTowne Landing. Can be tricky to find if you are not a local, but well worth the effort. It is a state park with tons of open fields for the kids to run wild in, a small "zoo" populated with animals the original settlers would have come across, a restored ship from Elizabethan times (no way would I have crossed the ocean in that thing!!!!), and very often historical reenactments. Last time we were there, there were a cobbler and a blacksmith. Other times, there have been women cooking, spinning, etc. charlestowne.org is their site
Not sure what the significance of Ashley River Road is

, so can't help you there.
Charleston is known for its amazing restaurants. Be warned, the ones downtown can be very $$$$. If you are into seafood, we make it a point to eat at the Charleston Crab House every trip. They have a couple of locations; we usually go to the one on Wappoo Creek, which was recently made famous on Flip this House on A&E. We went back Tuesday night for the first time since the renovation, and the food is still great. You can eat on the dock and watch the boats go by (and watch them raise the drawbridge a time or two). Again, not somewhere too touristy. charlestoncrabhouse.com is their site.
Enjoy your trip!