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Along NM 570 north of the village of Pilar, is a lovely section of the gorge that is very easily accessible. NM 570 winds along the banks of the Rio Grande at the base of the gorge, past several picnic areas, campgrounds, and trailheads where you can get up-close to the river, enjoy the scenery, and (in season) watch the rafters float past.
Perhaps the most accessible attraction in the monument, this steel arch bridge spans the Rio Grande Gorge a dizzying 650 feet above the river. There are observation platforms at each end of the bridge and a walkway on both sides of the road to allow pedestrians to cross the bridge, although it's definitely not for the acrophobic. A rest area is located on the west end of the bridge, with parking and restrooms.
Near the entrance to Orilla Verde Recreation Area, the visitor center has exhibits about the gorge and monument, with rangers on hand to answer questions.
Information about the park, with rangers on-hand to answer questions.
At the southern end of the Wild Rivers area, this point atop the gorge rim overlooks the confluence of the Rio Grande and the Red River, with good views over the plateau and of the distant mountains. A short interpretive trail runs from the parking lot to the rim itself, making it an easy place to hop out of the car and get a quick look.