Livorno-Pisa-Firenze are mostly served by regional (commuter) trains and most are 2 class only. In either case, no advance tickets. Civitavecchia to Rome is again mostly regional (commuter) trains with very little other than 2 class. Thus, no advance tickets.
If it's crowded, so be it. Regional trains are never sold out - a ticket buys you the 'right-of-passage' and does not guarantee a seat. I had stood the entire 3-4 hour trip from Pisa/Rome because there were no seats. Relax, smile, enjoy it, and make some new friends from Italy.
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