I'm not sure how the system works in the USA, but I imagine that it is similar to Canada, if you get your Prospector's License it is very easy (and cheap) to stake a claim on Crown (government) lands. Each claim is 400m x 400m (or 1500'x1500' depending on where you are exactly) and you can stake up to 49 contiguous claims per block. This gives you certain rights while exploring, such as erecting tents, cabin, etc, but you must also "work" the land, which means you have to do a certain amount of assessment work and file it each year to keep the claims in good standing, this is usually done by charging out your own time for things such as mapping, line cutting, sampling, panning, etc. There is also money available in the form of grants and other government incentives to get people out there exploring.
Prospecting is a great way to get out and spend some quality time in the bush, it can easily turn into a life in the woods (I used to spend eleven months of the year in the bush doing this sort of thing, and I did this for years). If this sort of thing appeals to you I would suggest taking a prospectors course, these are quite often offered for free (or almost free) by the local Ministry of Mines. In the USA contact the closest US Geological Survey office for more info.