Just based on driving portions of it, I'd avoid biking on US 11 until you get into Lexington where portions are signed for bikes. A quick search (start - Staunton, VA, keyword Lexington) at ridewithgps.com turns up at least 2 possibilities:
http://ridewithgps.com/routes/1621604 and
http://ridewithgps.com/routes/719840 Just looking at Google Maps, State Routes 252 and 39 would get you most of the way. Bicycle Route 76 goes through Lexington (I've seen signs in town) I think from Waynesboro. The other bike-mapping sites (mapmyride, bikely, garmin, strava) may have other routes folks have planned or ridden. I'd try contacting bike shops in nearby communities - Shennandoah Bikes in Harrisonburg (a bit north) may have suggestions. This link
http://www.lexingtonvirginia.com/cycling.htm may be useful.