Here is the definitive guide for renting a car at Civitavecchia after a cruise. I used Hertz which was efficient and convenient. I don't know where the other companies are located.
After docking and collecting luggage, the port provides a shuttle to the main bus depot. (You cannot walk in the port area). This drive took about 10-15 minutes and luggage was no problem. It was loaded by the driver. On arriving there, I was directed to a bus at the very end near the road which goes to the train station. Leaves every 10 or 15 minutes. Luggage has to be carried inside like airport shuttles and it was a bit tight but the bus was full and everybody managed. It cost 2 euros pp. Hertz is about a 2 minute walk down the road form the train station. I left my wife at the bus stop; walked to Hertz and collected the reserved car. It was all very efficient. Then went to get the luggage. Note that I paid about 35EU per day for a Fiat 500Lr reserved through AAA which fitted 2 and luggage but will be tight for more. I bough super cover for peace of mind and that was 45EU/day!!