This is my experience on how to get to Civitavecchia train station arriving from a cruise and carrying bags by your own.
(In case you you are considering to book a transfer directly from the ship you don't need to read this. Your transfer will stop along the ship at the terminal)
We arrived at CV cruise port on last Aug 10.
Ships may dock in any of several terminals. The cruise line don't previously knows which. In our case it was the furthest one.
As is not allowed to passengers to walk along the port, Celebrity provided us free shuttle buses from the ship to a small square just after the port exit.
The ways we figured out to proceed to CV train station (only one exists) are:
1 - Celebrity offered a shared transfer by van. Cost USD 25 per person. (we considered it too expensive).
2 - Taxi - there are taxis available there and the cost is around 15 to 20 Euros.
3 - Walk by yourself - some TA reviews say it is doable. But DON'T do that. Even though the distance to the station is just 1 Km, the path is terrible to be done with bags, Full of cracks, street crossings and blocked pathways by restaurant's tables and chairs.
4 - The last thing is one that nobody tells you and in my opinion is the best one.
Just take the public bus "Linea Speciale - Terminal Crociere - Stazione". Only 2 Euros per person.
It departs exactly from the ship's shuttle drop off point (impossible not to see it). Runs every 10 min. in a 5 to 8 minutes ride. Stops inside the station. Plenty of space for lugagge..
Our time line on disembarkation day was:
At 7:00AM entered the long disembarkation line inside the ship, claimed our bags, boarded the shuttle, got to the bus stop, took the bus. We arrived at the station at 8:20AM. Our train was booked for 9:16AM to Roma Termini.
So it worked perfectly.
Hope it helps.