Bring your passport, but also have a copy of your return flight itinerary, proof of accommodation, and proof that you have some money in an international ATM-networked bank handy. It is extremely rare for any immigration officer to ask for these, but according to the law they can refuse you a tourist visa if you cannot provide them.
Present your passport to the immigration officer for stamping. You will get a Schengen tourist visa, which is good for 90 days in Italy, or anywhere in the Schengen area.
Extend your visa past 90 days by leaving the Schengen area, and then returning to get a new stamp and another 90 days. This is getting harder as the EU expands, but Italy still has a neighbor outside the EU: Albania. This can be done by air, or by ferry from Ancona, Brindisi and Bari.
Extending your visa within Italy is another option, but it is not recommended. Italian bureaucracy is legendary for its ability to take an eternity to accomplish anything. At a minimum, be prepared to prove you have sufficient means to support yourself in Italy, proof of accommodation and proof of health insurance. You'll also need to make multiple visits to the Immigration Office.
