Rome to Civitavecchia transportation

Regardless of travel means, it takes about 1:15 to get to the city, and then you have to get (say) from the train station to the port's free shuttle bus station to get to the ship. You were too vague about assigned hours and month of the year to get the best advice. It would also be useful to know oldest and youngest age and average luggage weight. Does cost figure into "best?" The city bus to the port is half-hourly and can be crowded. Can each passenger climb bus steps with their luggage?

14 people is a special case, but the local (non-express) train is so cheap that it's still the cheapest. But the station is a sweaty walk to the port, even if you are fit. If there are people who might get left behind on the Termini platform or who can't put their own luggage away on the train, I'd consider hiring a 20-person bus. Your hotel likely has a ground travel partner because this issue comes up so much. Email the concierge for a quote.

Are you returning to Civitavecchia?

Our ages are 11 to 72, and all would have one small or medium case and a backpack.
We would be travelling early August.
It would be nice to travel together by train, or small bus. We wish to plan in advance to avoid confusion in Civitaveccha.
We are staying at an apartment in Rome. without a concierge to enquire about transport.
Are we best with an agent to arrange this?

We are not returning to Civitavecchia

I would say your apartment host might know somebody, and without personal experience (though I will probably be using them in May) with a company like Rome Cabs or Rome Shuttle Limousine, they might be best to check on prices.