During the "Gilded Age" in American history, thousands of immigrants came to what they believed was the land of opportunity. However, there were as many reasons for immigrating to the America as there were types of people who made the trip. The one unifying factor is that most immigrants came to America because they believed it offered something they currently did not have.
In the film Far and Away Joseph Donnelly came to America because he wanted his own land (along with the fact he wanted to avoid "execution"). In Ireland they believed "A man is nothing without land" and that "Land is a man's very own soul." Shannon Christie came to America because she believed America to be the land of modern people. She wanted a place of her own where she could have horses and could get away from all the social restrictions her family had in Ireland.
The Christie parents stayed in America to start over and have a new life. Daniel Christie at least also wanted to become part fabled "wild west." Others like Steven Chase came because someone they knew or loved was in America.
Many of the other "real" immigrants came simply looking for that better way of life. They wanted things like a job where they could make more money, or a farm where they could own their individual land. Others were forced out of their home countries and to America by overpopulation, crop failure, famine, violence, or industrial depression. (textbook. p. 539 top left) Those people that were forced out may not have really wanted to come to America; it is just all their other opinions had run out. America was the only real choice they had left.
Many lesser reasons also aided in the immigration during the Gilded Age of American history. For instance the invention and widespread use of large, swift, oceangoing steamships cut the time required for the Atlantic voyager from three months to often less than three weeks. This widespread use also brought about competition for passengers that greatly reduced the cost of ocean passage from Europe to America. (textbook. p. 538 bottom right)
However, not everyone had the same goals once they got to America. Some immigrants never even intended to permanently stay in America. "Young Chinese and Italian males often journeyed to American shores simply to earn enough money to return home and buy land or set themselves up in business." (textbook. p. 541 top right)
Most immigrants who came to America experienced widespread urbanization. I think very few immigrants actually made it out west like Joseph Donnelly and Shannon Christie did in the film Far and Away. The majority of immigrants settled near where they first landed, like in New York. Some of the luckier ones who arrived with a little money saved up might have gone on farther to Chicago, Milwaukee, or even the made it out to the prairies beyond. (textbook. p. 540 bottom left)
The artisans and craftsman with some technical skill sometimes did much better than the "normal" lower class immigrants. The lucky ones were able to find jobs that put their technical skills to work. The unlucky skilled workers more and more found their jobs being replaced by machinery, with their tradition skills no longer needed.
The labor conditions for unskilled workers were very often dangerous, and paid very low wages. Not only was the equipment they worked with dangerous and accident prone, they were also frequently required to work in environments where their health was severely affected.
The living conditions in the urban areas where most immigrants settled were also terrible. The tenement buildings, as they were called, were often were overcrowded. In these slums sanitation was practically unheard of. Garbage lying in the streets was very common as was the disease that often ravaged these buildings. The poverty of most of the tenants living in these buildings prevented them from finding a better place to live.
As one of the characters in Far and Away put it, "Unless you're rich, you're nobody." I do believe the prospect was there for immigrants to enjoy "the land of opportunity" but I also believe most were not able to take advantage of these opportunities. They were very often tied to their class both mentally and economically.
Probably the majority of the immigrants living in the poorest conditions had no chance to move on to better conditions. However, even for the ones that could have moved up, they may not have wanted to. Moving on to better conditions might have required moving out of their little "sub-town" where all the people would have been of the same nationality. Also factors like not knowing English would hold back some immigrants.
So yes, I do believe America was the Land of Opportunity for many immigrants. However, for most immigrants they came to realize that it was also a far-off dream. America had the opportunities but many of those opportunities that the immigrants originally came looking for were not available as readily or to the quantity as they had first hoped.
There are two quotes that I think pretty much sum it up. To realize the opportunities it took the hard work and the desire to realize them. It also took a measure of luck.
"The poorest immigrant comes here with the purpose and ambition to better himself and, given half a chance, might be reasonably expected to make the most of it." (source book. p. 75 mid right)
"Immigrants who got ahead in the late nineteenth century went from rags to respectability, not rags to riches" (textbook. p. 524 mid right)