The "American desert" can refer to an absolutely huge portion of the continent, and many, many different ecosystems. Its a big enough area of land that it is like saying I like hiking in Europe.

I'm partial to Utah's canyonlands, and a lot of the adjacent pinyon forest (still desert) in Northern New Mexico and Colorado, but that's just where I grew up. The Saguaro forests in southern Arizona are also amazing.

If you've never been to the desert in America, a good plan would be to fly to LA, and drive to the Grand Canyon. You will pass through a number of very different desert ecosystems.