Rome has a terrific underground railway system that will get you near most tourist destinations. The Roman Metro is designed in an X shape with line A and line B (plus a new branch of the B line, called B1, which goes from the "Bologna" stop to "Conca d'Oro"). Both lines cross at the city's central station, which is called Roma Termini; in front of it, there's also an important bus terminus.

Rome has also an extensive network of trams and buses that will come in handy: it's a big city so trying to see everything by foot is simply not feasible. The whole metro, tram, bus, and part of the train network (comprising the Roma-Lido, Roma-Giardinetti and part of the Roma-Viterbo railways) is managed by ATAC, whose site comes with a handy route planner.

If you will be in Rome for an extended period of time, buy a seven day Metro and bus pass (called CIS) for  24 €. For shorter stays, get a day pass for 6 EURO. An individual ticket is 1.50 € and is valid for 100 minutes. Tickets and Passes fares are here.

The 'Roma Pass' is another option; it costs € 36.00 and gives you 3 days of free bus and Metro travel plus free and discounted museum/monument admissions. Alternatively, there is also a 'Roma Pass 48 hours' for € 28.00. A real bargain.  http://www.romapass.it/

You can obviously catch other trains from the Termini station: for example, you can transfer to the Frecciarossa train in order to travel to Florence for the day. The Metro maps are easy to navigate: beware at rush hours that the trains can get very crowded and you'll most likely be pushed into and out of the trains. This is also the time where pick pockets make a killing - so keep your wits about you!