All of the suggestions by Stanton and RF are recommended although you really should see Stagecoach too. It is a great film and did more than any other to elevate the western as a serious genre at the time.
As for the series westerns....well I am probably alone here for finding them often entertaining and charming. These films really are of a different time and were aimed largely at a juvenile audience so they are much gentler and simpler than the sort of fare most of us enjoy in our movies today. I guess my soft spot for them was born of many happy hours spent as a child at the 'Saturday morning pictures' we used to have here. The films shown at these weekly kids events were often very old serials and b pictures from the 30s, 40s and fifties and I still remember them fondly. Should you decide to give them a go then I would recommend films featuring the following actors:
Johnny Mack Brown
Lash La Rue
The Three Mesquiteers featured various actors in this trio and changed over the years. At different times these included John Wayne and the aforementioned Ken Maynard among others.
As I said before, they are simple stuff and usually ended with the hero sorting out the bad guys with his fists rather than his guns and although there was usually a love interest of sorts the horse is more likely to get kissed than the girl. Not to everyone's taste but they can be fun if you approach them in the right way.
N.B. I would not recommend any of the 'singing cowboy' films. Although most of the above tried their arm at warbling the odd tune (even John Wayne) for the most part this was left to stars like Roy Rogers and Gene Autry. Both of which were hugely popular in their day but pretty hard to take now even by the most forgiving of us.