Howard Hawks, who made more good movies in different genres than any other film maker, once said that it was impossible to make a Western without John Wayne. The essential Hawks/Wayne collaborations are the aforementioned Rio Bravo and Red River.
As far as John Ford and John Wayne go, well it is hard to overstate the impact of their collaboration. Together they cemented the place of the Western, its images and tropes, into the collective mythological memory of the nation. The Cavalry Trilogy you allude to is, of course, essential viewing, but the best of the bunch --because they recognize and question their role in the myth-making process-- are The Man Who Shot Liberty Valance and The Searchers.
And yes, there was a time for me fresh out of grad school where I watched waaay too many movies. About 1000 in two years. No fooling.