If you want to venture on the train;

After arriving at FCO, head for the FCO Train station. You can buy ALL your train tickets at the Trenitalia counter - or use one of the kiosks. From there, ride the FM1 to Roma Trastevere - cost currently is €8 per person. Then, ride a Regionale train from Roma Trastevere to Civitavecchia - cost is now €4,10 per person. Total cost to you for three people is €36,30.

The time from FCO airport to Roma Trasteve is about 25 minutes. And the time to Civitavecchia from Trastevere is about 50-65 minutes, depending on which train you get on. The trains at FCO usually come about every 15 minutes (non-weekend and non-holiday). The trains at Trastevere to Civitavecchia are about every 30 minutes. So I would allot 2 hours for the trip time.

The article posted by Linda can give you some insight into the journey from Rome out to Civitavecchia. It's quite easy, although not nearly as fast as a shuttle... but it's about 1/5 the cost!