I get asked this a lot, so I'll tweet this out to everyone in case it helps.
Nobody does things exactly the same way, but I'll tell you what has worked for me. This will be Jewish-specific, but some of it is universal.
Start by finding your family in the 1940 census, the latest that's available.
Which are surprisingly available.
You can do most of this for free at familysearch.org.
You also want to be looking for birth, marriage and death records, to fill in maiden names.
Once you know that your family was Rosenberg from Proskurov, you can search Yad Vashem for that specifically.
But the most important thing to know is that TONS of info is out there. There's a misconception it's all lost or destroyed. Not true at all.
Oh, and nobody's name was changed at Ellis Island. But you all know that already, right? 😜
If your family says in the census they came from "Russia," that can mean anything from Estonia to Ukraine and everything in between. Sometimes even what's now Poland.