... well actually, it isn't a simple or easy matter to determine which newspaper was/is the oldest newspaper in the West, mostly because it is difficult to establish what geographic location constitutes "the West," or "the Old West." Depending on where you think the West started, there are several contenders. Based on a wealth of colorful history of newspapers and journalism in the Old West, there are a lot of interesting stories to be told.
If you accept "west of the Mississippi" as your chief criteria for what you mean by "the West," a worthy contender is the "Arkansas Democrat-Gazette," the newspaper of record for the state of Arkansas. It is located in Little Rock, Arkansas. The Democrat-Gazette wins the title because one of its ancestor papers, the "Arkansas Gazette," was founded in 1819.
But if you feel that Arkansas, though west of the Mississippi, really isn't traditionally a part of "the Old West," then you might want to consider the "Santa Fe New Mexican." The New Mexican lays claim in its masthead to the title of "The West's Oldest Newspaper, Founded 1849." A little digging on the New Mexican's website reveals it was first published in 1849, and that the first edition consisted of two pages in English and two pages in Spanish. That stands to reason, since Santa Fe had long been part of Mexico and the population in 1849 would probably have been more Mexican than "Anglo." (New Mexico became a U.S. territory following the Mexican American War, in 1848. At the time, "New Mexico" designated all the region between Texas and California.)
In addition to these two newspapers, there were many others with relatively old initial publication dates and respectable histories. In Denver, "The Rocky Mountain News," under the entrepreneurial leadership of William Byers, beat it's rival newspaper into print by a scant 20 minutes in April 1859, making it Denver's and Colorado's first newspaper. The "Rocky" as it eventually became known was a major force for shaping Colorado Territory and Colorado State politics with its often brash and never bashful editorial stands taken by Byers and his backers. Sadly, the Rocky succumbed to the loss of readership and advertising experienced by many modern newspapers. It printed its last issue in February 2009. (As of the time of this article, the final Rocky Mountain News edition was available on the Internet. How's that for irony, since the Internet was a major contributor to the death of the fine old publication?)
Then there's the historical claim made by the "Portland Oregonian" in Portland, Oregon. It seems to claim it is the oldest West Coast newspaper, with this reference online in "The Oregon Encyclopedia": "The 'Oregonian,' the oldest newspaper in continuous production west of Salt Lake City, Utah, began as the 'Weekly Oregonian' on December 4, 1850." Another online site of archived newspapers makes passing reference to the "Oregonian" with clear reference to it as "The longest running newspaper on the West Coast ...." Yet there are numerous newspapers that Internet searches would turn up in California with first-publication dates before 1850. Obviously the "Oregonian" has a claim to longevity as a West Coast paper -- but perhaps not the "oldest," only the oldest "in continuous production."
Contenders for the oldest newspaper in the West are many, with no clear winner because of the wide variety of publications and the very broad definition of "the West" or "the Old West" fans of the West will accept. But it is clear that newspapers sprung up early and often all over the American West. Pioneers and prospectors alike were interested in what was going on near and far. They were always hungry from news "back East" and loved reading accounts of the latest gold strike, cattle drive, or new settlement close to their new homes in the West.
This love of knowledge and hunger for news drove enterprising men and women to record the myriad names, places, events, and faces that give us the view of the West which we have today.