It's 9 Euros per person to take the train from Civitecchia to the Roma train station and from there you can get anywhere. If you find a group willing to share a van on your schedule it may be more convenient and not too overly priced. However, if you plan on staying on in Rome I would advise the train. I lied in Europe for 3 1/2 years and the train system is second to none.
You still have to get from the pier to the train station in Civitiavecchia, plus you have to haul your liggage on the train.
>> The original post inquired about getting from Civitavecchia to the Leonardo da Vinci Internatoinal Airport (FCO) in Fiumiciino. I think that the cruise line's transfers are the easiest solution.
>> Most of the cruise lines also offer transfers between Civitiavecchia and a hotel in Roma. Hotel packages generally include the corresponding transfer, but it's also possible to buy the transfer separately.
Note that the cruise line's transfers operate to or from the pier, so there's no issue of connecting transportation between the pier and the train station, or between the pier and a bus depot, or whatever, in Civitavecchia. The cruise line's transfers also have arrangements for luggage because they know that cruise passengers will have a lot of it.