The largest tribe in the United States in the Navajo (Dine) and they live in the four corners (Where Utah, Colorado, New Mexico, and Arizona meet).
Arizona and New Mexico have other famous tribes, including the Apache and Od'ham. And in the cities around these areas, you find relatively large populations of native Americans (University of Arizona, though all the way down in Tucson, has a respectable but small group of Navajo students.)
Living on the Navajo reservation can be interesting and depressing, I don't know about other ones. Flagstaff is where you go to shop, and sometimes due laundry (so you'll meet a lot there).
Washington state has reservations as well, and Seattle will have some native Americans.
Oklahoma used to be a big hub, many tribes being forced there, but it's not very native American compared to the states I've mentioned.
I noticed other people said Alaska. Officially, these are Inuit people.
There is no single Native American way of life. Just as Europe has different cultures, from Basques to Gypsies, Or Asia varies from Japanese to Mongolian, so too native Americans are very different (some are raiders, others farmers, others fishers).
But Arizona or New Mexico is probably the most "romantic" place for those who want to see "traditional" culture like pow wows etc.