Buy "Stagecoach," directed in 1939 by John Ford. This was the film that made Wayne a star in his role as the Ringo Kid. The first and perhaps best of the Wayne/Ford collaborations, it's ranked by the American Film Institute as No. 9 on its list of the 10 greatest westerns.
For one of the Duke's war films, get "Sands of Iwo Jima" (1950). Wayne was nominated for an Academy Award for his portrayal of Sgt. John Stryker. The film was also nominated for best screenplay.
Buy "The Quiet Man" (1952), which is neither a western nor a war film. Wayne played troubled boxer Sean Thornton, who returns to his ancestral Ireland after killing an opponent in the ring. Afraid of his own power, he refuses to defend himself when encountering small-town cruelty. Another John Ford film and one of many Wayne films featuring Maureen O'Hara as his leading lady.
Pick up "The Searchers" (1956), another Ford film, with Wayne and a young Jeffrey Hunter searching for a very young Natalie Wood, who was kidnapped as a child by Comanches. Wayne's character, Ethan Edwards, is a complex portrayal; uncle to the girl, he becomes increasingly obsessive as the story progresses.
Buy "Rio Bravo" (1959), perhaps the best of the 1950s/1960s westerns with Wayne in a senior statesman's role. Directed by Howard Hawks, it also has a great performance by Dean Martin as Dude, a drunken deputy who is drying out.
Get "True Grit" (1969), for which Wayne won his sole Academy Award as best actor. Famous for the line "Fill your hands, you son of a bitch!" (He bellows it merrily at a character who calls him fat.)
Buy, finally, "The Shootist," Wayne's farewell performance. He made the film after the removal of one lung and several ribs, and plays an old gunfighter dying of cancer. It was also Jimmy Stewart's last western.