If you're interested in Montana history and the history of the region, I suggest Montana: A History of Two Centuries by Malone, Roeder and Lang. It's been revised several times since it was first published in 1976, and it's still the most common general history of the state. You're also looking at a period in which Theodore Roosevelt was present, so books like Rough Rider in the White House or other Rooseveltian works will give you some idea as well.

The late 1880s are a fascinating time in Montana's history. The arrival of railroads into the territory starting in 1880 spurred a railroad-building boom that by the end of the decade had connected Montana from stem to stern. The buffalo had all but vanished by the start of the decade, victims of overhunting, but they were already replaced by cattle and sheep in their millions. The 1880s was the era of the open range, with the territory divided into districts drawn on a map. American Indian reservations were frequently encroached upon by these massive herds, but resistance was futile.

The railroads made it possible for herds of hundreds of thousands to exist. Easy shipment (in refrigerated railcars, a new innovation) meant riches were available for outside investors. There were books and pamphlets encouraging people to invest in the beef boom, and they did. Of course, the winter of 1886-1887 was disastrous ─ killing perhaps 60 percent of the territory's cattle ─ but the industry recovered quickly. Smaller operators survived the winter better than larger ones, and a spike in beef prices (courtesy of a shortage in supply), helped them out. Speculators were shaken out of the market at this time. The tough winter also spurred a huge increase in hay acreage (but that would not take hold until the 1890s).

The late 1880s are also the period of the huge quartz gold and silver boom. The 1860s were the era of the placer boom, when individual prospectors or small groups of prospectors could collect gold with not much more than a pan and a sluice box. By the 1880s, that easy gold was gone. The arrival of the railroads allowed heavy equipment to be brought in affordably, which allowed underground mining to begin in a big way. This was a pattern that would repeat across the West: Gold would be discovered by placer miners, some of whom would profit ─ but the big money would come later, once outside investors brought in heavy equipment.

In Montana, this "corporate mining," as History of Two Centuries refers to it, "paved the way for Montana statehood in 1889."

With regard to lifestyle, it would vary greatly depending on your career and your location, but it wasn't dissimilar to places across the western United States, Canada and Alaska during this period. Electricity would certainly be available in larger towns and cities along the railroad, but there was no centralized grid or rural service, except in rare instances. Rural electricity would not come to most places until the New Deal of the 1930s. Telegraph service reached Montana in 1866, but I'm not aware of any telephone service in the territory during the time in question. Water service was local, and the only reliable long-distance transportation was the railroad. Surface roads were all but nonexistent.

Company towns near the big silver, copper and gold mines were just getting started in the late 1880s, but they would soon blossom into profusion. Hard-rock mining was dangerous, and unionization would not really get going until the 1890s and later. The Populist Party, for example, would begin its sharp rise until after statehood.

History of Two Centuries has a pretty good description of life on the open range, but you'd do well to consult some of the Rooseveltian works for more color. This comes from page 168:

Most cowboys were young, probably in their early twenties, and energetic. Although our folklore fails to say so, they were often either African American or Mexican. Especially when on the trail or during roundups, cowboys worked long days with little sleep. They labored under trying conditions, facing searing heat and bitter cold, drenching rains or snows and choking dust, kicking horses and nervous cattle. For this, they received wages lower than what most workers received in settled areas, thirty to forty-five dollars per month, plus keep. The keep left something to be desired, for the camp cook fed them a steady diet of beef, bacon, beans, bread, and sometimes canned fruit. Canned tomatoes were a luxury and canned milk a miracle.

Cowboys relieved the monotony of their lives with practical jokes and wry humor. They cut loose on infrequent trips to such cow towns as Miles City, Chinook, or Lewistown, where they could find liquor and beer and such exotic foods as eggs, celery, and oysters. Saloons, the cowhands' social centers, offered not only drink but also entertainment and the companionship of prostitutes. In Miles City, the epitome of cow towns, the Cosmopolitan Theater or the Gray Mule Saloon provided popular shows of all kinds. While squandering their pay and raising hell, the cowboys frequently got out of hand and perturbed local residents. According to Nannie Alderson, an astute observer, the residents of Miles City considered cowboys a wild and undesirable lot. "Nice people in Miles City," she recalled, "would as soon have thought of inviting a rattlesnake into their homes as a cowboy."