Roanoke and Blacksburg are a decent place to start overnight trips, the land is just split between plain forest and the postcard hikes. A lot of the national forest around there allows dispersed camping unless a sign says otherwise, so you are looking for a backcountry night rather than a numbered pad. Permits and reserved sites show up more in wilderness areas and developed campgrounds, so that answer changes with the exact trailhead. McAfee Knob and Dragon's Tooth are gorgeous and also a rough first classroom with a full pack. A short in-and-out, where you can turn around if the weight feels wrong, teaches the overnight part without turning it into a sufferfest. Are you hoping to hike in a mile or two and camp, or do you want a longer ridge once the shelter and the food bag make sense?