Everyone needs to carry identification in Italy, this is normally in the form of an Identity card, these are issued by your local council (comune) Despite what the official site says, that you have to be an Italian Citizen, that’s not true, they are issued to anyone resident in Italy. There is two types, one for citizens that allows you to travel in Europe with it, and one to Italian residents that are not citizens that does not allow you to use it for travel in Europe.
When you open a bank account, apply for utilities etc, you will need ID, the CIE is the easiest open. Many banks for example won’t open an account unless you have an Identity card, and yes people often say tehy can accept a passport, but actually it’s up to the bank what they accept, not the customer.
Applying for it is easy, you go to your comune with your passport or driving licence, and your tax code (codice fiscale), plus a passport sized photo. If you are an Italian Citizen and live abroad, you apply via the consulate that covers your area or country. Normally takes 10 minutes or so to apply, and your card can be collected or sent to your address in 6 working days.
There is a fee to pay, this does vary, the card itself has a fee set by the Government, but comunes will charge secretarial fees on top, this means that for most the card will cost around €25. Your local comune will confirm the price, often on the website.
Official Advice is here, ignore references to Citizen, as explained residents are also entitled to it. https://www.cartaidentita.interno.gov.it/en/request-cie/release-and-renew-in-italy/
English (UK) 
Italiano 
Français 
Español 
Deutsch