At Civitavecchia you take the free shuttle bus operated by the port [don't pay for the one offered by Celebrity] to the port gates. Then you walk along the coast - keeping the sea on your right - for about 10mins until you reach the train station. Trains run frequently into Rome and take up to an hour depending how many stops. The fare is €9 return. Very easy to do and much cheaper than the shore excursion offered by the ship. But do remember the Italian tendency to go on strike at the drop of a hat! Use this site to find train times: http://www.trenitalia.com/en/index.htm.
By the way, there have been some major last minute price drops on this cruise - if the price of your grade room has dropped call your TA and they will adjust the price down accordingly for you and refund you the difference. Friends of mine have just got a Sky Suite for under £700pp!!
The train stops in rome at various metro stattions making it easy to get around Rome. Train busy in morning into Rome but generally seats available. Worth being independant and much cheaper.
Remember your train ticket also gives you free travel on Rome city buses and subway. There are two trains an hou.
We have done this trip its easy and cheap. Look on the trentalia site on the web it gives you times and prices and the trains are lovely double decked. Celebrity might be running a shuttle service into the nearest town where the trains depart if not get a taxi it isn't far. Or you can walk it's not to far. The train has several stops in rome make sure that you get off right at the end into the central station.