What did it mean to be an immigrant in America from 1800s to 1950s? What kind of expectations would an Irish or Italian migrant have when they arrived in Newyork?

There's an amazing documentary by Ric Burns called New York: A Documentary Film on the history of New York City from its founding until now. It's in 8 parts and in total about 17 hours long. If you're genuinely interested in the history of NYC it's worth it to watch the whole thing, it's all up on youtube. Otherwise you can check out one or two episodes pertaining to what you're looking for.