The book They Came in Ships by John Phillip Colletta has a flowchart. He suggests starting with the following:
- Your ancestor's full, real name
- the approximate age at arrival
- the approximate date of arrival
From there he suggests different types of searches, depending on the time period, and the other information you may have, such as the port of entry, the name of the ship, your ancestor's country of origin, etc.
He suggests looking for the answers to the above three questions in:
- oral family tradition
- family documents [such as letters, passports, Bible records]
- civil and religious records [including naturalization records, census records, marriage and burial records]
- published genealogies and local histories
Overviews of the immigration process can be found at BYU Broadcasting and at NARA (National Archives and Records Administration, aka The US National Archives). Check their pages Clues in Census Records 1850-1940, and Clues in Census Records, 1790-1840 to see what you might learn from a careful examination of the US Federal Census.
If your ancestor was naturalized, you may be able to find the arrival date in the naturalization records. The process of finding those also depends on the time period, because of changes in the naturalization laws. See the National Archives article "Where can I find Naturalization Records?".
If your ancestor is female, I recommend the article "Women and Naturalization, ca. 1802-1940" and "Women and Naturalization, ca. 1802-1940, Part 2" from The (US) National Archives' Prologue Magazine.
In some cases women can be found in passenger lists under their maiden names, so try that as well as her married name, if you know it.
I highly recommend Joe Beine's Emigration and Immigration Records and Resources:
and Stephen P. Morse's One-Step Web Pages (specialized forms for searching passenger lists, census records, etc.).
Other things to consider -- it helps to trace your ancestor's life backwards in time to narrow the time frame for a possible arrival. Make a timeline of what you know and how you know it. Be aware of where other relatives might live -- if your ancestor is on a recent passenger list, look for the column 'person going to meet in the US' to see if you recognize that person. You'll want to be able to recognize your ancestor's entry in the passenger list and be confident that you have the right person.
People rarely emigrated just once and completely by themselves. Some families went back and forth more than once -- many families were part of a chain of migration events involving multiple people from the same family or same community. Searching for the names of siblings or children or parents can find family groups traveling together -- useful if your ancestor's name happens to be indexed badly or was entirely wrong.