For transportation into Rome from Leonardo da Vinci International Airport (FCO), the Leonardo Express train is a widely-recommended non-stop service between FCO and Rome Termini (the main station). Trains leave every half hour, and the trip takes 32 minutes.
Rome has an extensive underground railway system that will get visitors to several major attractions. The Roman Metro has three lines - A, B and C. The metro system is reasonably-priced and quick, and metro maps are easy to navigate.
Rome also has trams and buses that visitors are well-advised to use; as it's a large city, trying to see everything by walking is not possible.