I am told the train service runs right into Rome at a fraction of the cost of an organised tour. Sorry have no details of where when and how much but you can find that out on the interne.
Civitavecchia train station is about a 10min walk from where the port shuttle bus drops you off. There are two trains an hour to Rome and the cost is 9 euro, which includes free travel on Rome local bus and underground.
The train stops near St Peters (San Pietro) where you can take the 64 bus into the square, which runs about every 5mins. For train details http://www.trenitalia.com/en/index.htm.
Last time we visited Rome some passengers chose rail as the cheap alternative, however, whilst they were in Rome, railway workers threw a lightening strike and a very expensive taxi got them back to the ship only in the nick of time. Bewar.