Yes, it is easy to take the train to Civitavecchia. Inexpensive, yes it is. Hard, long walk to to port, no it is not.
Yes, there are stairs if your train does not stop at track 1. Most do not. (1) Don't try to carry your bags down the stairs (unless they are small) - roll then down slowly, one stair at a time. (2) Going up, pull them up slowly, one stair at a time. Takes a bit longer but there is not back strain.
The walk to the port entrance from the station is not that bad. Slight downhill, then it levels out. It has never taken us more than 10-12 minutes, with luggage.
Take a hard look at your luggage. If your bags are in the 30-36 inch range, they are TOO big! You are not packing for a semester away from home! If a single bag is close to 50 pounds, you are overpacked!! Two smaller bags are much easier to deal with. Most people take way too much luggage.
Take a hard look at the wheels on your bags. If they are those little 1 to 1¼ plastic jobs; put that suitcase away and get a new one. Those little wheels will not survive the cobblestone areas in Italy (or Europe).
Radiance of the Seas - Nov 2006
Legend of the Seas - Jun 2007
Costa Fortuna - Mar 2008
Legend of the Seas - Oct 2008
Celebrity Constellation - July 2009
MSC Orchestra - Jan 2009
Navigator of the Seas - March 2009
MSC Poesia - November 2009
Jewel of the Seas - June 2010