The Greenway is wide in most places and directional signs with mileage estimates are placed at all intersections, as seen here. In most places, the pathway is away from the road and winds through lovely forest, though the pathway does parallel the road closely as you approach the western section of the Historic Village. I camped in the campground and biked to the Hermit's Rest Road three times and out to Hermit's Rest each time. Traffic on the road is light since buses and park vehicles are the main users, and the buses have bike racks if you need a lift back.
The 7.6 mile Hermit's Rest Road is closed to private vehicles (except those with handicap stickers) from March through November, but it is open to bikes and makes a wonderful bike ride. The views are spectacular and you get a good workout on the roller-coaster, up and down hills. Of course, you can rest as often as you wish at the numerous overlook points. Remember to carry and drink plenty of water as you drink in the views. The photo below is the start of the Hermit's Rest Road.
An example of the views from Hermit's Rest Road. The next photos are examples of the views from the Greenway and Hermit's Rest Road. You can see visitors at overlooks in both of the next photos.
A number of the overlooks feature views of the Colorado River far below (as seen here) -- something you cannot glimpse from the viewpoints in the Historic South Rim Village. If you get very lucky, you might even catch sight of a raft down on the river.
The Tusayan Greenway is also open now, traveling 6.5 miles from just outside the park to the Visitor Center.