B.I.R.G. tickets

You can select a train that stops at Roma S Pietro station and walk to the Vatican from there instead of training all the way to Roma Termini (main train station)which is past the Vatican. Cost is about 4.50€ from Civitavecchia. There are about two trains per hour in the morning that will stop at S. Pietro. For train schedules/prices, enter a date within 7 days (preferrably the day of the week you will be traveling).
http://www.trenitalia.com/cms/v/index.jsp?vgnextoid=ad1ce14114bc9110VgnVCM10000080a3e90aRCRD

Forgot to address the BIRG. I don't see that it's an option anymore. A one day ticket to use in Rome is the BIG (4€), but it wouldn't include Civitavecchia.
http://www.atac.roma.it/index.asp?p=19&i=14&r=677

There is a 3 day ticket BTI for 11€. If you plan to be in Rome at least 3 days, I would get the Roma Pass which would also give you free admission to two sites.
http://www.romapass.it/p.aspx?l=en&tid=2

The BIRG still exists. You need one for five zones, and it will cost 9 Euro.

The BIRG is not a Trenitalia option so you will not see it on the train site... it's actually part of the COTRAL system (METREBUS LAZIO System) - see this link: http://www.cotralspa.it/ENG/biglietti_abbonamenti_4.asp

As far as how the train system works, try these links:

Riding Trains in Italy - http://www.roninrome.com/2009/03/10/riding-the-trains-in-italy/

Civitavecchia to Rome by Train - http://www.roninrome.com/2009/02/28/cruising-come-to-rome-for-the-day-2/

BIRG used to be listed on the ATAC website, but they recently updated and I don't see it listed anymore. It may be there somewhere, but they didn't do an English version or they aren't finished with the update.

There are also COTRAL buses between Civitavecchia and Rome, but I think that the train to S. Pietro Station works much better to visit the Vatican.

Here's the Metrebus Lazio info for BIRG, BTR and CIRS.
http://www.atac.roma.it/index.asp?p=19&i=14&r=678

The BIRG ticket is valid on Ferrovie dello Stato (Trenitalia) trains, but only on the "regionale" ("R") commuter trains, and NOT on the long-distance trains (AV, ES-Fast, ES, ES-City, IC, ICN, EC, ECN, EXP). The "R" trains are the only trains which stop at Roma S. Pietro Station. You must buy the BIRG ticket for the travel zone of interest to you, and you must be careful not to "stray" beyond that zone of travel. The ticket must be punched the first time you board a train or bus, and its valid until midnight (unless you buy a BIRG ticket for a longer period).

Marlamap,

We are planning to do the same as you in December. Arrive in Civitavecchia port and then visit the Vatican 1st and then check out the rest of the sites that we can prior to leaving port @ 8pm.

I was wondering what you ended up doing and if you had any advice or recommendations.