Its very easy. Take the port shuttle bus to the port gates and from there walk a bit further along with the sea on your right to get to the train station, about a 10 min walk. The train fare last year was €9 return which includes use of public transport in Rome. Dont forget to get your ticket validated (date stamped) on the yellow machine on the platform. There are two trains an hour. For trains times see http://www.trenitalia.com/cms/v/index.jsp?vgnextoid=ad1ce14114bc9110VgnVCM10000080a3e90aRCRD
If you want the Vatican get off at St Peters (San Pietro) before central Rome. You cant miss it as you see the dome on your left. Its about a 15 min walk from the station or you can just jump on the bus (No.64 I think) at the station that runs every 5mins and get off at the second stop, walk under the underpass and you are there. From St Peters you can also take the tourist hop on hop off bus around Rome. Coming back dont leave it until the very last train in case of cancellations etc.
We have travelled from Civitavecchia to Rome by rail a number of times. Some cruise ships dock some way from the port entrance but shuttle buses are normally available to take you to there. From there, turn right along the front and the station is about half a mile on the left. There are frequent services to Rome - some faster and more expensive than others. There are two stations in Rome - one for the Vatican ( I think it is St Peters (San Pietro) but not absolutely sure) and the main station (Stazione Termini) which is a bit further on. You can walk to the Vatican from its station but I would suggest you get a map which you might get from your cruise line or at the station. Alternatively, there are maps available on the internet. From the Vatican, you can walk to the other sites such as Castel Sant'Angelo and then on to the Pantheon etc. It is a fair walk but well worth it. We have always walked so cannot advise on buses etc. From the main station, you can get to the Coloseum, Roman Forum etc. Again a fair walk. You could take a taxi but Rome traffic can be very busy!!
The trains can get very busy. We usually travel first class (prima classe) which is not usually vastly more expensive. A word of warning - remember to validate your ticket before boarding the train or you will be fined - there are yellow machines for this on all platforms.
Good luck. We are cruising to Civitvecchia end of October and will again be making our own way to Rome!
Agree with the other comments just make sure you buy the right ticket as some of the cheaper fares do not cover all the trains and you will get fined onboard.
You might consider going onto the Vatican web site and taking a tour organised from there. It will give you an earlier entrance, priority viewing and a much better experience than trying to DIY. The crowds and queues even in November can still be enormous. ... Neil.
Very easy to DIY - Try google and 'cruise ports call boards' - the first result will give you all the info under the Europe section
Just get off the ship and its about a ten minute walk to the railway station it cost about 8 euros and gets you there in just over an hour, make sure you get back to the ship for the departure as the ship will not wait for you, if your late . Get off at the Vatican City station.