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To locate information regarding the arrival of your relatives to this country, you might try searching Passenger Lists. If you believe they entered the United States through the port of New York, you can Ancestry.com for the digitized lists from 1820 through 1957.
As for additional records, you could request a search through the U.S. Citizenship and Immigration Services Genealogy Program to determine their citizenship status. New York City directories, recently digitized by the New York Public Library could list addresses and sometimes occupations. The National Archives holds records related to the U.S. Federal Government, so the records can be limited when seeking personal / family information. You could try reaching out to historical societies in the city where your relatives lived, the New York Public Library's Genealogy Division, or the New York State Archives.
Good luck with your research!