I grew up in the 70's when Clint was king. I think he's great, but John Wayne was my western hero. My favorites include:
The searchers (probably his finest acting, totally unlike most of his other films) Red River, another great western.
She wore a yellow ribbon (watch out for another great western actor in there, Ben Johnson, a real life cowboy and rodeo champion. Later to win an oscar in the last picture show, also in The wild Bunch)
Rio Bravo, a Howard Hawks classic. Watch out for stumpy played by the great walter brennon.
The man who shot liberty valance. Made in black and white. starring wayne and james stewart.
Stagecoach: wayne's big break. Made in 1939, it launched wayne in to the big time. Before stagecoach Wayne had mad dozens of b western churned out in a few weeks, without getting his big break. Although, in 1930 he had his first starring role in the big trail, he had to wait another 9 years for super stardom.
Many of the John Ford directed westerns feature the same actors. These were known as the john ford rolling stock company. Many appeared in wayne's films as well. For example, ward Bond, Ben Johnson, Bruce cabot (king kong) John Agar, Harry Carey Jn
Typical john wayne westerns worth a watch include:The war wagon: the comancheros (great music from elmer bernstein, composer of the magnificent seven ) Chisholm (full of regular John Wayne sidekicks) Rio lobo is another typical wayne western in the mould of rio bravo
The same off-screen names also appear time and time again in wayne films eg James edward grant, waynes favourate writer. the composer elmer bernstein, directors Howard Hawks, Henry Hathaway, Andrew V Mclaglen, and of course his mentor John Ford.
Not John Wayne, but Bad day at black rock with spencer tracy is a modern day (late 1940's early 1950's) western and well worth a watch.