I am going to agree with Pat on US-58 / Mouth of Wilson / NC Bike Route 4.
From Damascus you can either follow the Virginia Creeper Trail to Whitetop and join US-58 there, or I would suggest joining it at Green Cove, a mile before Whitetop and it's a little closer to US-58. The Trans-Am also leaves Damascus on US-58 and you can just follow it from there as well. After about 15 miles the Trans-Am will continue straight and US-58 will turn right so follow US-58. It will then past the turns for Green Cove and Whitetop before descending down to Mouth of Wilson, VA.
After passing Mouth of Wilson, US-58 will parallel the New River and VA-93 will turn to the right. Take this turn across the river, it is obvious with the bridge there. Immediately after crossing the river you will enter NC and this is the western terminus of NC Bike Route 4. NC Bike Route 4 runs east and west across the state along the northern border counties with Va. NC bike routes have signs, but signs can be missing. Pat provided the link to order the maps. If you send me an email I can send you pdf scans I have made of my maps of the route.
I would follow NC Route 4 to Danbury, NC / Hanging Rock State Park. Here is a link to a cue sheet the North Carolina Bicycle Club has from the Greensboro Amtrak station to the park and back as a round trip. I would give that a try to get into Greensboro.http://ncbikeclub.net/cue_sheets/hanging_rock.html
If you are really aiming for Raleigh, and don't really need to go to Greensboro and just through that in your note as an aiming point then I would just stay on NC Route 4 until it intersected NC Route 1 north of Oxford, NC and the Kerr Lake area and then follow Bike Route 1 into Raleigh.
If you need to go to Greensboro and then continue on to Raleigh then you can leave Greensboro on Alamance Church Road which will intersect NC Bike Route 2 outside of town. NC Bike Route 2 will take you to Raleigh after going through Chapel Hill.
Hope this helps.