Westerns pre-80s were just a vehicle for all the other genres of the time. There were kitchen table dramas, romances, cultural-political examinations/commentaries, detective stories, revenge stories, bright lights big city stories. It would be more reasonable to lump Star Wars and Gattaca together as "two common examples of SciFi" than to willy-nilly lump any movies with a horse in them as a "Western".
All Westerns share a similar stereotypical setting: They are set at a time before the pre-eminence of the automobile (City Slickers isn't a "Western") and in a place west of the Mississippi. That said, there are, very broadly, two kinds of the Western: First, there's the Mythos Western, the default. It has heroes not far divorced from those of the serials of the 20s and 30s. There are standard heroes with a close relationship to law and order, as well as anti-heroes who nevertheless land on the right side of morality. Such law that exists is embodied in a few men (and fewer women) who stand up to enforce it - either with or without a badge. So, Revenge Stories are pretty common. But so is the story of the hero who stands up for the Right even though no one thanks him for it.
But, at least as early as the 40s, a new kind of Western began to appear that was created for people who grew up on Mythos Westerns. The Anti-Mythos. They were complex people, often on the wrong side of the argument. These movies turned a mirror on the Mythos heroes to consider their problematic facets. They often considered what a true hero is. But to appreciate this second kind of Western, you need understand the feel of the Mythos kind. There's no hard recipe for the latter. You have to stew in them for a while. So my list starts with the Mythos stories and then moves on to the ones calling out and riffing on the Mythos stories.
1. True Grit (Coen Bros) - Since this is a getting-started list, the Coen's will ease you into it gently.
2. Tombstone - In many ways, this is an unremarkable Mythos Western but Val Kilmer's performance is stand-out.
3. True Grit (John Wayne) - Now your ready to take on John Wayne's performance in this movie. It's a revenge story. Robert Duvall, is Lucky Ned Pepper, the boss villain, and this is one of the two movies where Dennis Hopper dies in John Wayne's arms. Also, Glenn Campbell and Kim Darby.
4. Gunsmoke (TV show) - Just start with "Bloody Hands", "Seven Hours to Dawn", and "The Jailor" (with Betty Davis). What people forget is that Gunsmoke was launched as an "adult" drama series in a Western setting.
5. Maverick (TV show) - The "Gun Shy" episode with James Garner's Maverick (there were four Mavericks in the series including James Bond's Roger Moore as "Cousin Beau"). The episode makes fun of the motif's of the Gunsmoke series, such as why everything in town seems to gravitate around the four major characters, the long-view street shoot-outs, etc. This is Anti-Mythos but we'll put it here since this episode is so enjoyable to watch after a fresh watching of Gunsmoke. Essentially, the entire Bret Maverick character uses the Mythos Western hero as a foil. Think of your classic Western Hero from the serials, such as Tom Mix, or think of Gary Cooper. Or better yet think of Errol Flynn's hero in Captain Blood: smooth, earnest, noble. Now imagine the Captain Blood comedy sidekick character, Honesty Nutall, got his own show. That's Maverick in a way. Bret Maverick is lazy, cynical, conniving, and not the sharpest tool in the shed. He's competent if he has to shoot it out and he's smarter than most of his marks when he plays cards. But he's not a tough cookie per se. If he wins a physical encounter, it is by employing some tactical advantage. Garner's trailer-residing detective character, James Rockford, is a modern continuation of Bret Maverick.
6. Stagecoach (John Wayne) - All I'll say is that it probably is not what you expected.
7. High Noon - Gary Cooper is a town sheriff. Some time ago, he busted an outlaw that terrorized the town. Now the outlaw is out of jail and he and his gang are coming back for revenge. The town collectively decides they don't want to help him. As I said, This is a common example hero in Mythos Western. The one man standing for the Law when no one else can or will, and no one cares if he does or not. This was remade as a snoozy SciFi movie, Outlands, starring Sean Connery.
8. 3:10 To Yuma (1957) - Same heroic theme as High Noon. A guy is offered the chance give up his life to protect money that isn't his. A Western The Narrow Margin.
9. Hombre - Same heroic theme as High Noon. Based on an Elmore Leonard novel, Paul Newman is a white man raised on an Apache reservation on a stage coach with really awful people. He finds himself constantly in the position of having to protect these people and the whole thing evolves into a kidnapping/ransom caper.
10. High Plains Drifter - In this movie, Clint Eastwood puts a twist on the theme from High Noon, 3:10 to Yuma, and Hombre. I won't give any details since they would be spoilers, but it deserves its R-rating and would probably never, ever be made in America today.
11. The Quick and the Dead - Sam Rami's straight Mythos Western. Gene Hackman, Leonardo DiCaprio, and Sharon Stone (as a dead-eye gunslinger out for revenge). It's comic-booky but very watchable.
12. The Man from Laramie - Jimmy Stewart revenge story. If almost anyone else had done this movie it would be pure Mythos Western. But Stewart can never portray that sort of character. It has hints of Lee Marvin's Point Blank.
13. Nevada Smith - Steve McQueen searches the West for the outlaws who murdered his parents. However, this story considers the cost of exacting that revenge. The Red Dead Revolver video games owe almost as much to this movie as they do Clint Eastwood movies.
14, 15, 16. Fist Full of Dollars Trilolgy - FFoD, For a Few Dollars More, and The Good, the Bad, and the Ugly. These are Mythos Westerns but the first one is a remake of Kurosawa's Yojimbo which was a retelling of Dashiell Hammett's Red Harvest. which was also adapted as the unremarkable Bruce Willis film, Last Man Standing. Maybe the Coen Brothers' Miller's Crossing owes a little to Hammet.
17. Magnificent Seven - Mythos Western remake of Kurosawa's The Seven Samarai.
Now you've got a solid grounding in the Mythos Western, so you're ready to see the Anti-Mythos movies that play against your established expectations.
18. Red River - Directed by Howard Hawks, John Wayne's character is subversive to the stereotypical Mythos Western hero and the movie turns the Revenge story on its ear. This is also considered a touchstone of gay Hollywood.
19. Shane - An honoring, deconstruction, and rejection of the heroes of the Western serials. Joey Starett is a stand-in for the kids in the theater seats when those movies played. The film considers "Who is a real hero? The icon of Saturday afternoon fantasy who descends on trouble at dawn and leaves at sunset? Or is it the guy who slugs it out everyday for a woman, his children, and his community?" One of the landmark fight scenes of cinema.
20. The Searchers - Directed by John Ford. This is John Ford's Unforgiven. He had directed dozens of movies where settlers killed Indians who flung themselves carelessly into bullets. This movie examines the racial bigotry that underlied those movies. It is interesting that John Wayne - who is considered the standard for the Mythos Western hero - has so often been involved in landmark movies that directly unfavorably comment on and subvert the Mythos.
20. The Man Who Shot Liberty Valance - John Ford reinterpreting the West again. Here John Wayne is the Old West gunslinger who takes Jimmy Stewart under his wing: an attorney devoted to bringing civilization and law to West. Again, "Who is the hero?" The guy who kills every bad guy he meets or the guy changes the whole game that allows bad guys to operate without limitation?"
21. Unforgiven - Directed by and starring Clint Eastwood. It deconstructs the Western (and perhaps the Dirty Harry movies) where all disputes are settled with guns and murder with little consequence, and revenge is mercilessly exacted despite the fact that in real life the details would be much more muddy. Tarantino and DiCaprio's recent Westerns and the Deadwood series have taken no education from Unforgiven. They are good in their way but they display no more comprehensive knowledge of the so-called "genre" than Jonah Hex comics.
Extra CreditThe Lone Ranger (2013, Disney) - Armie Hammer as the man with the silver bullet. Johnny Depp at his Indian friend Tonto. This movie faces an uphill battle because it can make some people of a certain worldview feel uncomfortable. Depp is an Anglo actor playing an Indian. Whether Tonto had the role of subservient side-kick or magical enabler, it was going to be a bit offensive. Depp sidesteps this by making his character insane, an object of contempt in his tribe, and maybe not even a human being at all but a sideshow manikin animated by the Great Myth of the West.
Zorro (TV Show) - I never got a chance to try watching this series with my kids when they were young enough to enjoy it. No streaming yet. But it might be worth it to try now. Find a grade-schooler and see if these stories have aged as well as I think they have. It stars Guy Williams, "Professor John Robinson" from Lost In Space.
Zorro, the Gay Blade - The benefit of a mild familiarity with the Zorro TV show is that it will add to the pleasure of watching this wonderful, wonderful film that is to Western serials what Young Frankenstein was to Horror.
5 Card Stud - A murder mystery starring Dean Martin, Robert Mitchum, Roddy McDowell, and - really worth seeing - Yaphet Kotto.