We're arriving by ferry in Civitavecchia and need to travel to Rome. With luggage, etc., what is the best mode to get there? Train? Private shuttle? Other?

You could book a private shuttle which will probably run around 140€. If you take the train, most are regional trains (about 2 per hour) and will cost 5€ (travel time about 1:05-1:15). There are a few fast trains (16€ - 50 minutes), but I wouldn't hang around just to catch one of those. Regional trains don't have reserved seating (not possible) while fast trains do.

It's not expensive> Is it cheaper than 140 euros and is that expensive vs 10 e for two on train + taxi from where that train lets you off?

Since the OP has not defined "best" who knows? Having done the regional train MY "best" would be a car service which picked me up, ON TIME, at the ferry dock and took me to my final destination.

For some, "best" and "cheapest" are not the same, obviously.

yeh a car service port to door is always best and to get to the train at Civitavecchia can be a long walk or short cab ride.

3 or 4 years ago, we used a company called "Rome Shuttle Limousine". The ride cost 110 Euros, and the car and driver were both excellent.

http://www.romeshuttlelimousine.com/

the amount of luggage makes a difference whether to take train or not - if you can easily manage luggage navigating Termini station can be manageable -with loads maybe not.

But those trains can take you to several other small stations that may be closer to your hotel.

Train is cheap but with luggage not always so easy. Easy to get taxis at Termini though you may have to walk a ways to exits to access them.

If can afford it shuttles are way to go IMO.

I recommend a private car service; one cautionary tale, though:
Civitavecchia implies a cruise ship. We disembarked
from Oceania & had reserved a car. However, we had forgotten to account for the disembarkation time 8:30 AM? & had reserved
Car for 10. Big mistake!