I stayed here last year for a week on our family vacation in a camper. They have tent sites, not all sites have power and water. I would reserve a site, it fills up fast at least the sites with power and water and the better views.
As far as hikes, one of the most prominent hikes is to hike the Deep Gap Trail (I think that is the name) that goes from the summit parking lot of Mt Mitchell to the 2nd, 3rd, and 4th highest peaks east of the Mississippi, Mt Mitchell would be the highest east of the Mississippi. There are some cool views, the trail basically runs on the ridge line of these high peaks. I have done a long day hike in there and back out sumitting those 3 or 4 peaks, it was the culminating hike of our week long family vacation, beautiful.
Another cool area to look into is the Pisgah National Forest near Linville, NC. There is an area there called the Linville Gorge and a mountain, Table Rock Mountain that are really scenic. There is a dirt road running the ridge line on the rim of the gorge and there are parking areas all along it with named trails going down into the gorge. This gorge is referred to as the 'Grand Canyon of the East'. There is an awesome viewing area of Table Rock Mountain then you can do any number of those trails going down into the gorge. There is fishing all along the river down in there, beautiful area. There is a video on youtube of some guys that backpacked in to camp and fish that shows off the area a little. I have not been down into the gorge just drove the dirt road and stopped at several overlooks, Table Rock in particular. It was so gorgeous in there I was just dying to drop down into there for a backpacking trip and plan to go back soon to do it.