There are indexes to ships passenger lists for most ports
for the time period after 1820. The only exception is New York.
You can get a list of film numbers (NARA & FHC) at
http://olivetreegenealogy.com/ships/nara_indexbyport.shtmlScroll down to the state you want and click on the link,
then find the film # you need, then decide whether to order
it in to a nearby FHC or have NARA do a lookup.
The staff of the National Archives will undertake a search
of the original records for a fee.
To find your nearest FHC you can go to
http://www.familysearch.org/Eng/Library/FHC/frameset_fhc.aspYou might also want to try naturalization records, if you
need to narrow your time span for looking. There are great
clues in the 1870 census! If your ancestor has a check mark
in the column "Male Citizens of the United States of
twenty-one years of age and upwards" you have a clue that
naturalization took place before 1870 :-)
To find out what questions were asked on various census
records see
http://allcensusrecords.com/usa/There is an explanation and Resource Guide to
Naturalization Records at
http://naturalizationrecords.com/ There are also many links to searchable online
naturalization records at
http://naturalizationrecords.com/usa/Choose your state of interest from the list at the bottom of
the page then click on ONLINE records.