Sometimes discussing specific cases can help understand the general techniques for researching Sicilian and Italian genealogical records. Below is a request from a reader, and my response. Although it deals with specific surnames, and towns in three different provinces, my responses apply to any search for ancestors. Simply replace the names of the persons and towns below with those of your forebears and their villages, and search the venues indicated for images of original records, where possible. Though many individuals now may be searched for on-line by entering only their names, not every name has been indexed, and often those that have, have been misspelled.
Note: after my correspondence with the reader, she revealed that the Augello’s and Frattalone’s were from Sommatino and Canicatti, and that the Naso’s were from Marineo. My response follows.
Images of original civil records of birth, marriage and death for Sommatino (in Caltanissetta Province) are available on microfilm for the years 1820 through 1910. They are available on line for the years 1866 - 1910 on three venues: they are free on the Mormon site familysearch.org and the Italian Archives site at http://bit.ly/ItalianRecordPortal; and they can also be viewed on the paid genealogy site, Ancestry.com. In addition, records for 1820 – 1865 are viewable on line at Mormon FamiilySearch Centers (FSCs) or participating public libraries.
From Ancestry.com, the 1874 birth record of Calogero Augello shows his parents' names as Ignazio Augello, age 34, son of the late Sebastiano, and Crocifissa Sciascia, age 22, daughter of the late Calogero. A margin note on the first page states that Calogero Augello married Concetta Frattalone on 21 September 1902 in Sommatino. At one of the on-line venues noted above you should be able to find their marriage record, which would give their parents' names and whether they were still living in 1902. If deceased, look for the parents’ death records, which will give their ages and their parents' names.
Concetta Frattalone's birth record will also be on line. It will give the age of her father, and you should be able to guess whether he was married after 1865 (men usually married at about age 24). If so, his marriage record will be on line. Once you work back to ancestors with records prior to 1866, you'll have to view them at a Mormon FamilySearch Center. To find a center near you, go to http://bit.ly/LocateFSCs (Aside: the late Supreme Court Justice Antonin Scalia's paternal ancestors were from Sommatino.)
Civil records are also available for Canicatti (in Agrigento Province) for the years 1866 - 1910 on Ancestry.com; they are also on the Italian Archives site, and are available for all the years 1820 - 1910 on family search.org. Again, for records prior to 1866, you must use a FamilySearch Center computer at an FSC. You cited some information you gleaned from transcribed records at familysearch.org, but wherever possible, you should forego the transcribed records and find the actual images of the original 'atti' (records) of births ('nascite'), marriages ('matrimoni') and deaths ('morti'). These records are in the Italian language, but they give much more detail than the transcribed summaries, which also often contain errors by the transcribers. To help in reading and interpreting the original records, I suggest a book like 'Discovering Your Italian Ancestors' by Lynn Nelson, and viewing of on-line tutorials like the one on familysearch.org at http://bit.ly/ItalianResearch
You originally said the Naso family was 'from Palermo'. Immigrants often gave the name of the province that they were from, and in Sicily and much of the mainland, each province has a capital city with the same name as the province, leading to confusion. When searching for records or seeking assistance in doing so, it's important to make the distinction. Since it turns out that the Naso family was from the town of Marineo, in Palermo Province, you can search for records from that town. Marineo, as well, has on-line civil records for 1820 through 1901.