To make long stories short:
Just take the FREE SHUTTLE BUS from your ship to the Port Entrance Area
When you arrive make a turn to the right to exit the port area. Follow the street
and following sign:
When you see on the left side the HOTEL DE VILLE change the walkside from the promenade
There you will find on the left side the railways terminal building:
When a ship is arrived (like with us more than one) there are long queues:
But we booked in advance via Internet http://www.trenitalia.com
and so we can just collect our tickets at this machine:
But do not forget to validate your tickets by pushing them in this cancellation machine:
If you want to go to Roma Termini you may use any train (incl. IC trains).
If you want to go to Roma S. Pietro (Vatican) you can only use Regional Trains. Some of them starts from outside platforms of 1, i. e. turn right and walk to the end of the hall on the right side.
Trains to S. Pietro need approx. 45 minutes.
Trains to Termini need approx. 50 minutes (IC) to a little bit more than 1 hour (Regional Train).