Best way to travel from Civitavecchia to central Rome and what will it cost?

There are 2 Regionale trains every hour, about 43 and 58 past the hour, with some slight variations. Cost is 5 euro. The 43 past the hour is a Veloce and takes 1 hour and 5 minutes, the 58 past the hour is a regular making more stops so it take 1 hour and 20 minutes. You'll arrive at Termini where I assume you will want a cab.
There are a few other trains during the day that cost more and are Express trains.
The station at Civitavecchia is likely to be a zoo with the cruise ship dumping a ton of people. I'm not sure if the station ticket machines accept cash, but it would save a lot of time if you could use one to buy your tickets.