ATAC, Rome's public transportation company, is quite efficient.
There are only two subway lines and they run from 5 AM to 11:30 PM. Line A, the red line, goes (more or less) east-west and stops near the Vatican (Ottaviano-Vaticano), Piazza Del Popolo (Flaminio),the Spanish Steps (Spagna), Piazza Barberini, Termini Station, and Piazza della Repubblica.
The Line B, the blue line, goes north-south and stops at the Tiburtina Station, Termini Station, the Colliseum (Colosseo), the Circus Maximus (Circo Massimo), the Pyramide (Piramide), San Paolo Basilica (San Paolo) and three stops at the EUR.
ATAC also has some trams (the Tram 8 that you can catch at Largo Argentina will take you Trastevere) and many, many buses! You can easily recognize the bus stops with their yellow posts. The numbers of the buses are on it. If you see a number and an owl, it is a night bus, a great service for all night lovers. They can bear the same number as a day bus but their itinerary may be slightly different.
Better check on ATAC's website (they can compute your itinerary for you) or with the driver.
A single ticket (BIT) is 1,00 Euro; a day ticket (BIG) is 4,00 Euros; a weekly ticket is (CIS) 16,00 Euros. You can buy your ticket at the machine in the subway station, in selected supermarkets and (important if you take the tram or bus) at tabacchi and bookstands.