When we visited Italy we spent the majority of our time walking and taking the train. We did take a couple buses, we rented a car for a few days to explore Tuscany, and took a taxi once.
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Planes, Trains, Buses, Cars & Taxis: How to Get From Here to There in Italy
The airport in Rome is actually about 30 miles from the center of the city. After getting off the plane and gathering your luggage the next order of business for us was to make our way to the Rome Termini or the main train terminal in Rome. To do this we took the Leonardo Express train to Rome. The train ride was about 30 minutes.
From the Leonardo Express we found ourselves in the Rome Termini. From here we purchased a five day train pass.
Note that tickets for both these trains are not for reserved seats. You can use the ticket at any time within the next several months, but the ticket must be validated just before travelling by punching the ticket in one of the little green and white oval machines dotted about the platform area. (Make sure it prints the time and date on the ticket.)
If you fail to validate the ticket you could be fined, as you are effectively trying to travel on an open ticket. You may find a long queue at the first ticket machine you see on the way to the trains, but there are numerous machines/booths and shops all selling tickets at the same price. There is also a travel agent across from the ticket booth, who will sell tickets to any destination in Italy, with no markup.