In the U.S., you can request these from the United States Citizenship and Immigration Services (USCIS) or the National Archives and Records Administration (NARA).
If you are in another country, write to the National Archives there.
If your ancestor never naturalized, you must get proof of this, which includes a letter from the government agency that issues the documents. Also, many death certificates will state where the decedent was born, and census records are helpful because they may also state citizenship status. The more proof you have, the better.
You will need to get the certified, long form versions of all direct-line vital records, and they must have an apostille from the Secretary of State of the state from which they were issued. An apostille is the legalization of a document for international use. Usually, all vital records for your direct line not from Italy must be translated into Italian. The naturalization papers do not need an apostille nor do they need to be translated. For New York City vital records, you need to request a Letter of Exemplification, which will allow you to get it certified by NYC, which will then allow you to get the apostille from New York State. Depending on where you live, you may also have to go through a similar process.
Finding and dealing with discrepancies on documents is the most frustrating part of this process. It is important that the dates and names are consistent. Because many immigrants did not read or write, they may have signed papers with misspelled names and wrong birth dates. Sometimes, their memory just failed. My grandfather’s naturalization documents contain errors in his birth date and the spelling of his surname. Naturalization documents cannot be amended. I am hoping that a letter from USCIS will be sufficient, which states that the three different spellings of my Grandfather’s name represent the same man. To deal with the discrepancy of the birth date, I requested a letter from his comune in Italy saying there was no one by his name born on the erroneous birth date. Some documents can be amended, others cannot. Amend those you can so that birth dates and names are consistent. If you cannot amend a record, get a letter from the issuing agency saying so.
Eventually, you will go to your appointment and, hopefully, everything will go well. The wait to hear the outcome can take a couple of months. Once you receive a letter from the Consulate saying you have been recognized, you will make an appointment to get your Italian passport.
When times were difficult, Italy lost many of its sons and daughters to other countries. In a generous spirit, Italy has opened her arms to the descendants of those who left. If you decide to embark on this journey, just remember that you are not alone. Here are some helpful websites:
For a database of Italian towns, go to ItalianSide towns database
Ellis Island has records of immigrant entry into the U.S. at www.ellisisland.org where you can search the records by name or by ship.
You must follow the guidelines posted by your particular Italian Consulate; however, in the citizenship section on the website of the Consulate in San Francisco, California (USA), there is a detailed Guide to Italian Citizenship by Descent that you can download. It explains a lot and you may find it helpful even if you are not applying through that Consulate.
I hope this helps explain a bit about the process and helps get you started on your own journey. Buona fortuna!