The Outsider, (1961) is a black and white movie about Ira Hayes, who helped raise the American flag at Iwo Jima, in WWII. It is interesting to see Tony Curtis, at times very good, and at times, awful. He bonds with blond, James Franciscus, after an initial harassment by the Franciscus character. When his buddy is killed, Hayes turns to alcohol to cope with his grief. He also sings, "Where Are You?" while in a drunken stupor. (I know! Even at that time, I thought it out of sync with the story)
Incidentally, the esteemed singer, Dinah Washington recorded that song, after the film came out.
I Heard the Owl Call My Name.(tv movie, 1973). Tom Courtenay in the lead. I love it! The title refers to Native belief that if an owl sounds a certain way, at a certain time, you will be dying soon. Beautiful story!
Navajo. (1952). Well-reviewed black and white documentary. The boy actor, Francis Kee Teller, received raves for his performance at the time
I personally enjoyed Pierce Brosnan, in Grey Owl (1999) White actors weren't approved at playing natives at the time. But the story explains his reasons.

Thunderheart

Little Big Man

Powwow Highway
Dead Man

Ernest Goes to Camp
Dead Man
The Last of the Mohicans
Skins (2002)

Not exactly a recommendation, but if you're looking for accurate portrayals of historic Native American life,
Soldier Blue (1970)
has something to offer.
It's not a good film. Dull for the most part, a poorly written and acted romantic comedy (kind of), but it stands out because of the final scene. The film suddenly shifts gears with an extended and brutally realistic depiction of an attack by the US Cavalry on a peaceful Native American village. I won't give details, except to say that it's still my most harrowing movie-watching experience to date, desperately upsetting. If you've read 'Bury My Heart at Wounded Knee', you'll know what to expect.

Dreamkeeper

Yep. Amazing. Another excellent Inuit movie to see is Nicholas Ray's
the savage innocents (1960)
.

Atanarjuat (The Fast Runner)
has got to be one of my favorite movies about aboriginal people. It was made with a mostly Inuit crew, and a totally Inuit cast. It's a great story. It sometimes feels like a Shakespeare Tragedy played out on the arctic tundra. It also takes place centuries before they had contact with Europeans.

Hate leads to anger.
Anger leads to the Dark Side.
This is a Haiku.

Windtalkers
(2002) Native American recruits in WW2 who use the Navajo language to send 'coded' messages.
"Barney SloaneThat's my new nameMy old one's a little more Italian."

Hello imdb11168,
have you seen "Blauvogel" (1979, former GDR)? Story takes place in the 18th century. Robin Jaeger starring in his only movie role. There's no version with English dubbing or subtitles, I'm afraid; but I think the story is easy to follow.
And I would like to mention "One Little Indian". Saw some screenshots only; I haven't seen the movie in full, for there is no release on region 2 dvd. Might be available on y**t*be, but I prefer to have the movie on dvd for my collection. (My only chance is some day they might make a "Best-of-James-Garner dvd box set"
Perhaps "The Education of Little Tree" might be worth watching for you, too?
Last not least - in case you're also interested in movies set in regions of America others than the territory belonging to the U-S.A. of today: Set in South America (Brazil, I guess), you might find "At Play on the Fields of The Lord" interesting. Has just another one-time-actor, Niilo Kivirinta
Kind regards
sprendlinger

More modern era films
The Exiles (1961)
Powwow Highway (1989)
Smoke Signals (1998)

Little Big Man (1970)
Buffalo Bill and the Indians, or Sitting Bull's History Lesson (1976)
Bury My Heart at Wounded Knee (2007)
not seen yet
Cheyenne Autumn (1964)
not seen yet
Broken Arrow (1950)
The New World (2005)
Cabeza de Vaca (1991)
Jeremiah Johnson (1972)
Dead Man (1995)
Meek's Cutoff (2010)

In addition to the already mentioned titles:

• Thunderheart (1992) - takes place in contemporary times though.
• Pocahontas (1995)

six picks..
western
ulzana's raid (1972)
cheyenne autumn (1964)
little big man (1970)
devil's doorway (1950)
run of the arrow (1957)
apache (1954)
colonial
the last of the mohicans (1992)
black robe (1991)
the new world (2005)
unconquered (1947)
seminole (1953)
mohawk (1956)

The Revenant?
Unfortunately, a lot of older films about Native Americans contained white actors playing natives. For example, a man called horse.
Also, please ditch the "indian" term as Native Americans are not from India.