So you want to visit the old west, do you?
You want wide open spaces. You want untamed wilderness.
New Mexico has the lowest population density of any state in the continental US.
There is a huge spectrum of terrain here. Most people think of New Mexico as a desert. We have mountain lakes, snow capped peaks in the middle of summer, green river valleys, hidden hot springs in pine forests where you can sit comfortable through a night and watch the constellations set. You can get a backpack, load a tent and a sleeping bag and hike to places where you can camp, fish, hike, swim, and stand on the highest peak for hudreds of miles in any direction and not see another person for weeks. It sounds like this is what you are looking for.
We also have volcanic mesas, barrel cactus, desert sand dunes, indian ruins, cattle ranches, and ghost towns, if you head south.
Colorado is also a beautiful place. You can visit the old mining towns, with their locations along hard cliffs. Go to Boulder/Denver area and see the Rockies, some of the most beautiful mountains anywhere. The camping is a bit crowded, though, since most of the best places have been made accessible they've been ruined.
If you're coming to the west, I would not neglect to visit Arizona and the Grand Canyon.
Since I live here, I'll link some images that feel very familiar to me.
Toas pueblo. Probably the most touristy place in New Mexico, other than Santa Fe, but this building is very representative of what inspires most architechture in Northern New Mexico
http://www.internettime.com/images/taospueblo.jpg
A picture of the Jemez River, where my friends and I used to go cliff diving and camping.
http://www.wallpaperdave.com/wjem03.jpg
Wide open spaces
http://www.chattanoogaphoto.org/images/Janis - White Rocks New Mexico.jpg
I've heard many times that you can't rent a car if you're under 25, but I've never tried. I don't recommend walking - things can be very spaced
BrotherAP