It could vary significantly based on why the town was founded. A lot of mining towns just happened to spring up around a discovered site of precious metals. Many of these boom towns would then disappear when the ore played out, and became ghost towns. Rapid City in South Dakota is an example of a boom town that survived past the boom.

Many other towns were planned communities. Someone would either need a collection of people at a certain site, or would view it as a financial opportunity. They would then acquire the land and sell lots, with an intentional plan. The railroads in particular made a ton of planned communities. In exchange for building the railroads the government would grant a certain amount of land on either side of the tracks to the companies. The trains themselves needed regular watering stations. Steam engines need water to turn into steam after all. Every few watering stations, there would be a full station. That would be the access point for everyone using the train line, passengers, mail, and commercial goods would all be loaded on and off at these full stations, and would serve the rural area for 20 miles in every direction.

The plot of the Mel Brooks movie Blazing Saddles actually played out several times in real life as the train companies tried to maximize their profits by controlling all the land near train stations. The Colorado town of Durango is an infamous case. The train was being put in up toward the town of Animas. And the train company tried to negotiate for land in town. The townspeople sensing a potential to make profit themselves were driving a hard bargain. So the company bought some farmland a couple miles away, and laid out their own town, Durango. Within a few years most of the population of Animas moved the couple miles down the river, to take advantage of the station.

Of course it's almost impossible to discuss he settling of the West without talking about Brigham Young and the Mormons. He directed the settlement of over 350 towns from Canada to Mexico and from Iowa to California.

The center of Mormon settlement initially was the Salt Lake Valley. Young would select a few dozen families and "call" them to go and settle a likely location. Rather than settling solely around Salt Lake City, he intentionally scattered groups far and wide so as to stake claims for the Mormon communities to all the best farming locals within 300+ miles of Salt Lake. He would also intentionally settle certain areas so as to gain access to certain commodities. For example Saint George and other communities in SW Utah were intentionally established to grow cotton, importing clothing being prohibitively expensive. Price and Carbon county was settled for coal mining. Cedar City and Iron county were settled for iron mining.

In a lot of ways Young resembled a video gamer playing Civilization. Oh I need iron? Better put a settlement here. They'll need food, I'll put a farming settlement nearby. Both those will need lumber, I'll put a small lumber town there. Young was a great planner. Salt Lake City is a very planned town. But he went far beyond that, Young planned the entire region. That's why Vegas is where it is. Young needed a fort and a watering hole between Southern California and Salt Lake. Vegas was the convenient location. He even tried founding some lead mining communities near Vegas, but they didn't pan out. And there are lots of similar stories.

So that's at least three of the ways some communities were founded. Some came about by happenstance (Rapid City). Some were intentionally founded by corporate especially rail interests (Cheyenne). And some were founded to serve greater communal needs (Las Vegas).

It should be pointed out that regardless of beginnings many of these settlements have changed from their foundings. Vegas was far more defined by the building of the Boulder Dam than by earlier Mormon farmers and miners. So it can be difficult to pick apart individual pasts without study. Durango may have been a railroad town, but it is far more shaped by mining. Each town has to be studied on its own. Broad strokes don't necessarily do the founding stories justice, like I've just given you.