I updated my page Your Guide to Finding Immigrant Origins. As usual, links had gone bad, I found a few new sites and removed a few that seemed less relevant. But the basic advice remains the same: do thorough research in the United States first. Your goal is to know the original spelling of the name and the town of origin before you start researching in the records of another country.
This page on my website references also two of my other web pages, which I’m afraid I’ve not yet checked for bad links. So be forewarned, but these pages do cover important steps in your search for immigrant ancestors.