Since the West is the largest and most diverse region of North America, there could have been any number of scenarios. In addition, saloons were one of the first businesses to open in a frontier town, but they also opened and closed throughout the history of that town, so there could be a great deal of difference between that first establishment and a business that opened ten or fifteen years later.
J. Ross Browne in his "A Peep at Washoe" describes merchants with just enough kegs of beer and bottles of whiskey to open a saloon in 1860 arriving in the newly-established Virginia City (then of Utah Territory; after 1861, Nevada Territory). The key to these first arrivals was first of all to arrive with the goods (which sounds obvious, but it was no easy accomplishment with poor roads, etc.). They would set up a tent and a blank of wood on two barrels and start selling their wares. The life of the saloon might be no longer than the time it took to sell everything that merchant brought. Restocking might entail his returning to his place of origin (in this case Sacramento of San Francisco would be likely) to obtain more goods and then to return to set up his saloon again. Or he might make an arrangement with a packer and then later with a teamster to bring in more product, and he could, then, sustain his saloon past the point of the exhaustion of his first inventory. Both scenarios occurred.
For the opening of saloons later in the life of a frontier town, it would be possible for an entrepreneur to rely on well-traveled and established trade routes. Goods flowed into frontier towns from throughout the world, so it would be a matter of what the saloonkeeper wanted to sell as he (or she!) decided on ways to make his business different from others. One thing a saloonkeeper might do is to establish a contractual relationship with a brewery so the saloon could offer beer on tap (as opposed to bottled only). Saloons offering fresh beer in this way frequently marketed themselves as "saloon and brewery", which did not mean they were like modern brew pubs, but rather that they had an arrangement with a nearby brewery (which might serve any number of other unrelated saloons).
Briefly, to answer the question of /u/Chernozem, financing of that first-on-the-scene saloon could take very little in up-front money. It could represent someone's own life savings that he hoped to amplify with outrageous charges being the first on the scene, or it could be a number of people throwing together the funds needed to accomplish this. In all likelihood it would not be a formal bank loan. Later, banks frequently loaned the money needed to purchase a building and set up a business. Our research in Virginia City indicates that many saloons lasted only 3 to 6 months, so it wasn't uncommon for a saloonkeeper to sell his business at least figuratively "lock, stock, and barrel" to the highest bidder, who might use a loan to purchase and then hope he could make more of a success of the location and stock than his predecessor had.
A good source for all this is Kelly Dixon's "Boomtown Saloons." Kelly excavated two of the four saloon we excavated in Virginia City. I summarize her findings and my additional research in my book Virginia City: Secrets of a Western Past. (2012) I hope this helps.
"Frontier Town" could be quite a broad category, and what was available in the Dakota Territory probably differed quite a bit from what was available in, say, West Texas, but there is one interesting example from archaeology that comes to mind that can shed light on what people in one location were drinking at a specific time in the opening of the frontier (1860's). Note that this example would indicate that saloons in this area were being stocked via shipment by steamboat from points south (New Orleans) and East (major cities on the East coast). The steamboat Bertrand, which sank on the Missouri River north of Omaha, NE, en route to the goldfields of Montana, was covered in mud and silt and the cargo preserved as a kind of time capsule of frontier life in the 1860's. On board was an enormous selection of booze, much of it really top end stuff imported from Europe and other places around the globe. A great deal of high end pre-packaged foodstuffs were found as well, indicating that these folks were eating and drinking better than what you might expect.
As listed on page 50 of this National Park Service archaeology report, alcohol found included: ale, bitters, "bourbon-whiskey cocktail", brandied cherries, brandy, "Brandy Cocktail", "gin cocktail", wine, and champagne. I'm not sure what the distinction is between say, brandy and "Brandy Cocktail", but it is so-listed in the manifest. The diversity of foodstuffs is pretty amazing too, including various kinds of nuts, sauces (including Worchestershire, tartar sauce and something called "London Club Sauce"), canned meats, various fruits... these are luxury items. Again, not representative of life on the entire frontier, which comprised a vast and shifting area over many decades, but this report provides a snapshot at consumption habits at a very specific time and place on the frontier. In summation, they were bringing stuff in from more established areas, and drinking and eating pretty well!
Source linked above: "The Steamboat Bertrand", Jerome Petsche, National Park Service, Department of the Interior (Washington D.C., 1974).
Edit: a couple spelling, added the source citation at the bottom.
A second, but related question: how would you have financed it?
This answer does not answer the question of food stock, but does speak to early stocking of alcohol. The earliest saloons were basically lean-tos and stocked with locally brewed, rough whisky made with raw alcohol, burnt sugar and a little chewing tobacco. This type of locally-brewed liquor took on such names as Tanglefoot, Forty-Rod, Tarantula Juice, Taos Lightning, Red Eye, and Coffin Varnish. Also popular was Cactus Wine, made from a mix of tequila and peyote tea, and Mule Skinner, made with whiskey and blackberry liquor. The house rotgut was often 100 proof, though it was sometimes cut by the barkeep with turpentine, ammonia, gun powder or cayenne. Saloons also served up volumes of beer, but in those days the beer was never ice cold, usually served at 55 to 65 degrees. Though the beer had a head, it wasn't sudsy as it is today. Patrons had to knock back the beer in a hurry before it got too warm or flat. Unfortunately, my sources do not explain the source of the beer.
Eventually, some saloons were able to drag in blocks of ice from nearby lakes. Old West historian Kathy Weiser cites the Laguna Vista Saloon in Eagle Nest, New Mexico, as an early example. In general, however, until the 1880s pasteurized or refrigerated beer was essentially unavailable and patrons were accustomed to room-temperature drink.
Others may speak to the shipping of beverages after the Gold Rush and other boom-town events, which falls outside the scope of my readings and which were much different than the above imbibes, including unheard of drinks like $10 champagne with breakfast.
Sources: Saloons of the Old West, Richard Erdoes; Great American Bars and Saloons, Kathy Weiser.