The order of things:
1. Emigration Records (on their way OUT to a new place)
Most important for Continental Europe - emigration indexes from Germany etc. Ancestry has many of these)
Also note that families from places like Austria often stopped in England for a layover and may be found in British lists - example
2. Passenger lists to North America
Main ports: U.S. New York, Philadelphia, Boston, Baltimore, New Orleans, Galveston, San Francisco
Main ports: Canada: Quebec City, Montreal, Halifax, Nova Scotia, St. John, New Brunswick, Vancouver, B.C., Victoria B.C.
After 1820ish period everyone was supposed to keep these lists in the U.S.
Look carefully ath others on the list; how big a ship was this? Is everyone from the same place?
If your ancestor came in through Canada later in the 19th century they may appear in border crossing records.
If they had relatives in Canada or lived in northern states you might also find them in the border crossings which are at FamilySearch and Ancestry
3. First Papers / Declaration of Intent
Naturalization process first established in 1790
Residency for first papers changed at various times and ranged from arrival to 14 years but most often it was around three years.
Watch for witnesses in these kinds of papers; they might be relatives or long standing friends vouching for their character.
If your immigrant ancestor came 1798-1828 they were supposed to go to the local county court where arrival was recorded in the minutes.
In some places, they kept an alien's registry (sometimes called Report and Registry). They were supposed to record their name, birthplace, age,
nation, where they planned to settle and occupation - example from Maryland
4. Naturalization Petition - the form you filed to get your naturalization certificate. It had to be filed within 3 years of your Declaration of Intent
5. Naturalization Certificate - all requirements had been met and the court swore you in and gave you your Certificate of Naturalization
Naturalization after 1906 asked much more specific questions and the process could include visas and alien registration etc.
6. Passports - if your family came in the late 19th century they may have maintained relationships with people back home - you should check the passport indexes (Ancestry) in case they ever returned for a visit.
7. Homestead - if your ancestor was foreign-born and filed homestead it should include immigration information. Only a fraction of these are digitized so far - mostly in Kansas and Nebraska - but go to the Ancestry card catalog and look for the database U.S., Homestead Records, 1863-1908
George Martin Thomas Moran Mary Crain