The first thing you do, is to pull copies of all their death and birth certificates, then pull all the birth and death certificates of their brothers and sisters(sometimes the information is different, or more complete on a sibling's certificate.
From the death certificates, get copies of the obituaries from the local paper of wherever anyone related/in the family died, to get even more names to get records on.
Next thing you do is to get the all the census records. The census records will tell you when they immigrated, where their parents were born, and their names.
After you get the birth and census records, you should have enough information to find the marraige and immigration records.