In Italy you must always dial the area code AND the telephone number, even if you are placing a call to someone in the same city.  If you call Italy from overseas, you should dial the country code (39), the area code (complete with the initial '0'), and the number.

All Italian residential (fixed) telephone numbers start with a "0", and have between 8 and 11 digits (area code+number). 

Cell (mobile) phone numbers all have 10 digits, starting with a "3". 

Italian toll free numbers start with an "8" and have normally 9 digits, e.g. 800 111 222 (the exception being a few numbers having only six digits, e.g. the Italian Post office info line is 803 160, the Italian Automobile Club roadside assistance is 803 116). To use those numbers, which are often toll free also if calling from cell phones, just dial the number as given, without any additional prefix.

If you have to make an international call to the U.S., dial: 001 + (area code) + (number). That includes if you plan to call an American cell phone number, regardless of where that cell phone may be physically located.  WARNING:  It may be impossible to call a 1-800 U.S. telephone number from Italy even if you're using an international phone card to call the U.S. 

If you have to make an International call to another European country (e.g. France),  dial: 0033 (or other country code) + (area code) + (number).

 Other option is to rent a cell phone before your departure, with a local line from Italy, you'll have the benefit of low calling rates, including incoming calls free of charge. You'll find several companies offering this kind of service, be aware of those that use call back since the voice quality is very poor.  Some good options are http://www.phonerentalusa.com, www.worldcell.com and www.planetfone.com

Pre-paid telephone cards to be used to call from payphones can be bought at electronics shops, tobacconists' (tabacchi), newsstands and bars. Public phones, however,  are becoming less and less widespread: as you will notice, Italians are addicted to mobile phones. You can rent a mobile at major airports, although many long term visitors choose to buy a cheap cell phone for less than 80 euro and a pay-as-you-go SIM card from one of the three cell phone providers (TIM, Vodafone and Wind).

A link to international phone codes

Italian directory assistance is available on www.paginebianche.it

Italian yellow pages are available at www.paginegialle.it

For people from the United States: If you choose to use the payphones in Italy, do yourself a favor, if you only speak english buy an AT&T or whatever calling card in the U.S.  Make sure it has lots of minutes, and use the 800 access number for Italy. ( You can get that by calling the company before you leave the states).   Telephone operators in Italy may not speak enough English to help you place a call.   

If you plan to travel for a long time in Europe and need a phone your best bet is probably buying a used tri-band phone on Ebay or some other site and then just purchasing a 20 euro sim card.