Hi there, first things first, there are TWO international airports serving Rome, Fiumicino (aka Leonardo da Vinci ) and Ciampino. You are probably at Fiumicino, but please check.
I'm unaware of any direct train from Civitavecchia, so for ease, take the train to Roma Termini and chang ethere. The journey takes about an hour depending on time & type of train.
This station also serves the airport, both with a direst service, and a local/Metro train both taking arounf 35-45 minutes.
If you intend using the local train, you can actually hop off at a couple of stations before Termini (Roma Ostiense, or Trastevere) as the line is shared, but it's not really worth the effort as you'd have to carry all your luggage over the bridges.