I would say, most likely, Montana and Wyoming. Also the Dakotas west of the Missouri River. The entire area is sparsely populated and even though the USA's western frontier was pretty much 'closed up' by about 1890, a lot of those old cultural vestiges remain.Hallo from Denmark
My son talks about, that he likes to experience real cowboy western indian culture, like in Back to the future 3. And maybe also a modern large city still with western culture.
Which state to visit?
Parts of Arizona and New Mexico, too, but with the rapid population growth there over the past few decades, not so much anymore.
Larger cities in northern Texas also try to maintain a lot of the old culture.
It is a fascinating era of history and the adventure and hardships of that time and place are the stuff of legend. Enjoy!