Here are my choices in order of preference:

Shane - a perfect western in every way.
My Darlin' Clementine (there has to be at least one by John Ford and one with Henry Fonda, so this one handles both requirements while being an awesome movie on its own right)
Lonesome Dove - future generations might move it up higher on the list. It improves with more views.
Fistful of Dollars (there has to be at least one by Sergio Leone and one with Clint so this one does double duty also)
High Noon (the music moves it up here)

Near misses - Unforgiven, Outlaw Josey Wales, Magnificent Seven, Silverado, Open Range.