Vermillion Cliffs National Monument, Arizona

I browsed maps and web sites a few years ago. From what I recall US89 and US89A skirt on the north and SE sides. I believe monument headquarters are on 89 west of Page, as is one campground. There's a trailhead from the campground to one of the canyon walks.

89A crosses the river on the first bridge (actually its later twin) across this part of the Colorado, and a side trip takes you to Lees Ferry, with camping and river access. It is also the lower trailhead of the main canyon trail (Paria Canyon)

Then there's a dirt connector on the west side of the monument( House Rock Valley Road). There are few rustic campgrounds on this, as well as trailheads to the canyon walk(s).

Overall this is much less accessible (by road) than National Parks.

http wrote:
Access
From Flagstaff, travel north on U.S. Highway 89 to 89A at the Bitter Springs turnoff. From Kanab, Utah, take U.S. Highway 89 to the east or 89A to the south through Fredonia and Jacob Lake. There are no paved roads within the Monument. While the House Rock Valley Road/BLM 1065 is a maintained dirt road it may be impassable when wet. Other roads on the Monument require a four-wheel drive-high clearance vehicle due to deep sand.

http://www.desertusa.com/Vermillion-Cliffs-MN/photos/VermilionCliffsMap.pdf

You can use Google Maps Streetview to preview the sights are along the paved highways.