In most U. S. western movies they depict army infantry as the government authority in those times. Were the marines only utilized for amphibious warfare alongside the navy?
In short, yes, the Marines were utilized for amphibious landings.
In a bit more length, the Marine Corps was declining. The U.S. Civil War, shortly before the Wild West time period (1865-1895) delegated the Marine Corps to blockade duty. They were poorly, if ever, utilized in battle for the Northern States.
Soon after the Civil War ended, the Navy (upon which the Marine Corps was attached), moved from sail boats to steam powered ships. This left Marines without a duty. Marine marksman were left without a place from which to shoot, as all the masts were replaced with smoke stacks.
Much of this time was spent in introspection on part of the Marine Brass (higher ups). The traditions of the Corps were defined in this era, and its duties were slightly broadened.
The Marine Corps had a few chances to use their skills abroad, rather than at home, during those years. First, in response to a native uprising in Taiwan (1867) that killed off the crew of an American ship. The Marines actually lost that one. One Marine died, and zero natives.
They made up for that loss in Korea (1871) when after a disastrous attempt at diplomatic overtures, Korea fired upon US ships. With no apology coming, the Marines took part in an attack that took over five forts, and left 243 Korean soldiers dead, and only 3 from US forces.
Marines were also involved in the fighting in the Samoan Civil War, though in no great numbers. Their value was proven by the 3 medals of honor earned there.
There were many smaller landings during that time, Japan, China, Egypt, Panama, Mexico.
But the only other one of historical note is the coup that overthrew the Queen of Hawaii. The conspirators used the possible danger to American citizens as an excuse to have the Marines sent in. The Marine Corps landed, and patrolled the streets, but were not required to fire any shots, or take over any buildings. Their mere presence was enough to force the royalists in capitulation.
The Marines wouldn't really come into their "fighting force" until the Spanish-American War (1898), at which point the "Wild West" had been tamed.
It may be of note, that in those years, the Marine motto Semper Fidelis (Always Faithful) was created, as was the Marine Corps Hymn, and the Eagle, Globe, and Anchor emblem they wear today.
I hope this has answered your question.
Wow thank you for a such an outstanding and well written answer!