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).
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.).