This is the starting point in the public records for you to look for family information that was previously unknown to you. It is often the easiest place to search.
If you find your family in the Federal and NY State Census, you will find a lot of information that can lead to other records. Often the Census asked for immigration status: year of arrival, citizenship status, year of naturalization, address, a list of the people living in the household, occupation. You will find the ages of all the family members and where they were born.
You will find some children in the 1920 Census but not in the 1930 Census. That means they moved out, married, or died between 1920 and 1930. With that piece of information you can try to search vital records to see if you can find a marriage or death certificate.
Knowing when and where they were naturalized may make it possible to locate their naturalization papers. Some of these are online now. Some will require that you visit a specific court house.
More to this in a later post.