John Wayne:
Stagecoach
Angel and the Badman
Red River
Three Godfathers
The Quiet Man
The Searchers
The Man Who Shot Liberty Valence
True Grit
I was wiping away tears so hadn't been able to post a response when I saw TH had chosen John Wayne next (what? Macho guys can wipe a tiny tear dabbed from the corner of the outside of the eye now a-days, as long as its quick and no snibbling).
Most of my 'must watch' list for John Wayne is his later films and for the following reasons:
1. Rio Bravo/ The Quiet Man Why these two for #1?
First of all, the quiet man is very John Ford who is on the top of my list for Duke directors. Hal Wallis is the other, for me personally.
Second, because when you put Ricky Nelson, Dean Martin, and Walter Brennan into a movie with John Wayne, you are making a classic and quintessential Western. I would absolutely grant that John Wayne's acting ability is better displayed in other movies, but that the other reason is that this one captured my "Duke" attention from a young age. It is the one that 'sold' me. For others a bit older, "Hondo" would have been (has been) their John Wayne induction movie. It is briefly a story about a washed up sheriff (Dean Martin) who has become the town drunk and allowed an ambitious bad man (Ed Asner) to take over his town. John Wayne has to not only sober up his old friend, but hold the town law together while doing so. My favorite John Wayne joke comes from this movie:
Duke fan 1: I bet John Wayne falls off his horse.
Duke fan 2: No way, John Wayne never falls off of his horse. I'll make that bet!
-Sure enough, a rifle is fired and John Wayne falls off his horse.
Duke fan 2: Here is your money.
Duke fan 1: No, I can't take your money, I've seen this movie before.
Duke fan 2: Me too. I didn't think he'd fall for that again!
If you like[d]
Rio Bravo, you'd appreciate
El Dorado which follows a similar drunk sheriff (Robert Mitchum) and town surrendered to an ambitious badman without scruples. that John Wayne and a young James Caan must fight while nursing the drunk sheriff back to health.
John Wayne made another good picture with Dean Martin called
The Sons of Katie Elder which is worth an honorable mention here too. When the mother of 4 sons dies, the sons set out to find what happened to their family ranch and money, against the dying wishes of their mother.
TH did a good job summarizing
the Quiet Man so I'll not add much more. The movie draws you into the Irish countryside and way of life of yesteryear in an incredibly nostalgic way. I would bank money this movie is the reason for many Irish vacations when it first came out. What It's A Wonderful Life does for Christmas, for me, the Quiet Man does for an Ireland Countryside way of life (and in that reflects all rural life and customs in a positive nostalgic light as well).
I have a hard time at this point deciding which route to go. John Wayne did mostly westerns but also stretched his roles (a far stretch when he played Genghis Khan) in movies such as The Quiet Man. So, I'm divided as to whether to go to his obvious other successful western movies like
True Grit,
the Cowboys,
The Undefeated,and
Big Jake (two more of my personal favorites of his westerns), or to mention
The Sea Chase, or
Blood Alley, or John Wayne's other war movies such as
The Sands of Iowa Jima, In Harm's Way, or the Green Berets which he made to help raise awareness and support for troops in the Vietnam Conflict.
John Wayne is best known and appreciated for his Westerns and war movies. He had mild success in his romantic leading man roles but most are unknown or forgotten by men.
Perhaps for the women who still love his films, he still fits the bill for the classic strong and silent type in those roles.
Gene Tierney as Laura Hunt
Dana Andrews as Det. Lt. Mark McPherson 
Clifton Webb as Waldo Lydecker
Vincent Price as Shelby Carpenter
Judith Anderson as Ann Treadwell
