Sorry there is a real difference between a National Park and a National Forest Wilderness area. Nation Park, Don't get off the trail. National Forest Wilderness area, foot, mules, and horses. No motorized equipment. Some also have posted no Bicycles in Wilderness areas. In a National Forest you can file a claim, if you wanted to, or just do "recreational prospecting" only going down one foot, unless they changed the laws again. For Recreational Prospecting there is no bond to be posted.
In the State of Arizona the first claim marker must be a rock pile 3 feet high with a map of the claim inside, or a 1.5 inch by 1.5 inch with it being 4 feet above the ground with the map of the claim posted on it(We used 6 foot 2x2's for walking sticks in case we wanted to file a claim). You then file your claim at the County seat for $5. But until you have Discovery anyone can file over you if they have Discovery. That was the law in 1980 when my brother and I stopped at the B of Mines in Tucson. We also got a copy of the filing rules at that time.
I think my Sister-in-law may still have the booklet, as my brother passed away back in 2005. Recap
If you are in the park area, be very careful of where you walk(Stay on the trail). If in the National Forest, go anywhere you want on foot in the Wilderness areas, it belongs to all US Citizens.