Virginia Dare bridge is fine. I just biked to the Outer Banks a few weeks ago, and I had no problems. I did not think it was fun ride, though. There's a wide shoulder/bike lane and even a pull off point near the apex where you can stop and enjoy the view. The view is mostly water. The bridge is 5 miles long, I think. It was initially kind of neat, but it just went on too long without much variation in the view.

I haven't biked it, but I think I was once told that the Manns Harbor Bridge had much less traffic, but also was not as pleasant when traffic was passing because there wasn't a good shoulder. I could be remembering that wrong, though, so take it with a grain of salt. Personally I have no issue with Virginia Dare even if traffic is heavy because there's so much room. Just watch for debris in the bike lane.

If you go south from Nags Head, you will eventually hit the Oregon Inlet Bridge, which I've heard is unpleasant to ride on. I've ridden the southernmost stretch of 12 from Ocracoke to Avon, and the road is mostly fine although sometimes the sand dunes intrude on the shoulder of the road. It's also got some monotonous stretches where there are dunes on either side of the road blocking your view of anything but the long, straight road.

Behind the Wright Brothers Memorial is a bike path that's part of a couple of signed bike routes: 1 and 3. It's a mixture of paths and side streets in that area. I follow Route 1 up to the Southern Shores area, into the back roads that parallel 12 (signs say the roads are for residents only, but I've never been hassled on my bike), and you can pop out on 12 going into Duck. Duck has bike lanes, but when vacation traffic is high, the bike lanes are an obstacle course of people walking (the bike lanes double as sidewalks), people pulling out of or in to the businesses, and people riding the wrong way. North of downtown Duck there is a bike path that's pleasant, and once you get to Currituck County, there is only a shoulder, but it wide enough to be comfortable, although once you get to Corolla, you might see some people riding against traffic again. I enjoy the ride to Corolla and back. With only one road, you can't really make a loop of it, but closer to Nags Head you can go bay side one way and ocean side the other.