Mark:
Great area to ride. If you are staying at Table Rock State Park, just imagine that everything north of 11 is very good stuff. You will especially want to head to the little resort communities of Highlands and Cashiers. GREAT biking roads, and neat little areas to hang at for lunch. If you are at Table Rock, try taking 11 south until you pass Lake Jocassee, and take the immediate right to head north. I think the road is 130 maybe. There was some road construction a few months ago when I was there, but it was no problem. Anyway you can head to both Highlands and Cashiers from there.
You would also want to explore north out of Table Rock. Take 11 north for just a ways until you hit 276. Take 276 north to Caesars Head State Park. Its been a long time since I went this way, but it is beautiful. You can keep north on 276 to Brevard. Brevard is another neat little place. Continue on through Brevard on 276 for a couple of miles until you hit the red light where 276 goes left into the Pisgah Forest. One of my favorite rides is taking 276 into the Pisgah Forest here on the Davidson River Road. You wind along the river for a ways, then start to climb on some nice twisty roads on up to the Blue Ridge Parkway. You could then take the Parkway south towards Smokey Mountain National Park, or you could get off the Parkway a little early and head south toward Franklin and circle back towards Highlands and back to Table Rock.
If you have more time, going north on the Parkway would take you up toward Asheville and on up toward Mt Mitchell, the highest mountain east of the Mississippi.
Of course, you are also pretty close to the north Georgia and east Tennessee mountains. I am not as familiar as some of the others here on those areas, but you are indeed close.
If you need one, there is a Triumph dealer very close to you in Greenville. Unfortunately, Triumphs seem to be an afterthought for them, as they mainly sale the Jap bikes. A great dealer is in south Asheville, Myers Triumph Ducati.
Have a great trip. If you need any detailed information, just let me know. I am very familiar with this part of SC and NC.
Mike