Get around

By bus
Buy bus tickets before boarding from corner stores and other shops. The payment system for most mass transit in Italy (trains, city buses, subway) is based on voluntary payment combined with sporadic enforcement. Specifically, you buy a ticket which can be used at any time (for that level of service, anyway) and when you use it you validate the ticket by sticking it into a machine that stamps a date on it. Once in a while (with varying frequency depending on the mode of transportation) someone will ask you for your ticket and if you don't have one you get a fine, and theoretically (sometimes happens) you can be asked to present to the Police for a formal report. Usually line enforcers aren't very condescending, especially in northern Italy. In almost every city there's a different pricing scheme, so check in advance ticket formulas and availability.
For tourist may be very convenient to buy daily (or multi-day) tickets that allow you to travel as much as you want in a single (or more) day. Every major city also has some type of City Card, a fixed-fee card allowing you to travel on local public transportation, visit a number of museums and giving you discounts on shops, hotels and restaurants.
Check for both these possibilities at local Tourist's Office or on city's website (which is often of the form www.comune.cityname.it as for example www.comune.roma.it).