I started out with a video because this is a topic that comes up repeatedly. There are some specific rules for starting research for an immigrant. The first and most important is that you start the research in the country of arrival not the country of departure. There is a caveat for this rule if you already know exactly where the immigrant lived in the country of departure. The reason for this rule is simple. Unless you know EXACTLY where an event occurred in your ancestor's life in the country of origin, you have no place to start looking for records. You also have no way to distinguish your ancestor from all of the other people with the same name. An illustration of this rule is simple. Do a Google search for the ancestor's name. I just did a search for one of my ancestors on Google and got 245,000 results.
Once you accept the idea that your search for the immigrant is really no different than doing research for any other ancestor who lived in the country of arrival, you will just continue to do basic research until you find a record identifying the exact location of an event the ancestor's life. I spent about 15 years looking for one immigrant ancestor until I finally found a record where he recorded the town he came from in Northern Ireland.
I am not saying that you might spend a long time finding the information but without that key piece, you really have no way to connect a person with someone in the "old country." Now there are exceptions. Occasionally, an ancestor has an almost unique name. The place where that name is found is a good indicator of a possible homeland. Sometimes, you can find information about the actual trip across the ocean or from one country to another. The place where the immigrant lived could be found in emigration records as well as immigration records and by connecting the date of arrival with the date of departure some people can be found.
One good example comes from doing Latin American research. Unique names are quite rare in Spanish speaking countries. I once had a friend whose name was José Martinez. He had seven sons and every one of them had the same name: José Martinez. Literally. However, by identifying the parish church where the person was baptized or married and give a fairly accurate date, you can find almost anyone.
Almost any genealogically significant record could give information identifying the necessary immigration information. The list is endless. You might find the precious information in a letter, a Bible, a journal, a baptismal record, a marriage record (as I did with my Northern Irish ancestor), or anywhere else. The idea is that careful, systematic research of all possible records sometimes leads to results.
If you want a good place to begin, I suggest the links featured in The Family History Guide. Here is a screenshot of where you might start if your ancestors came to the United States or the earlier colonies.
Another place to begin learning about immigration records is the FamilySearch.org Research Wiki. Here is the webpage for the United States Emigration and Immigration.
The more you learn, the more places you will think to look.