Columbus, Georgia would be a great destination for a bicycling vacation. The Columbus Riverwalk is a 22-mile paved path along the scenic Chattahoochee River. The path runs through historic Columbus, near the Space Science Center, and through the National Civil War Naval Museum at Port Columbus. This is a great place to start your tour with ample parking, bathrooms, and a great museum.
Callaway Gardens, also near Columbus, has bicycle trails. You can see most of the gardens from the seat of your bike!
The Silver Comet Trail is another rail-trail project just outside of Atlanta, running all the way to the Alabama border. You can ride your bikes across the old railroad trestle and through a tunnel in the mountain. Again the old railway is very flat, the surface is paved, and the only traffic will be other bikes, runners, and skaters. The town of Rockmart is a good starting location.
The Augusta Canal is a great location with lots of history and a lesson in hydrodynamics. This path follows the old canal towpath, so it is flat and easy to ride.
The town of Madison, Georgia is very bike friendly. This is the town that Sherman did not burn, so the old houses are beautiful and the narrow streets are great for biking.
Only a little further from home, travel down to Tallahassee and the ride the St. Marks Trail. This is a rail-trail project so the grade is very flat, the surface is paved, and there is no traffic.
Virginia has two nice trails with a crushed-gravel surface: theVirginia Creeper Trail, and the New River Trail. I have ridden both of these trails while vacationing in Virginia. A mountain bike with smooth-tread tires works very well for these two trails. The New River Trail is 52 miles long, and the Virginia Creeper is about 30 miles long, so a week-long vacation would be well-spent.
For more bicycle vacation ideas, see the Rails-to-Trails Conservancy