Rome to Lucca

Whether it would work for Florence to be visited as a day trip really depends on your interests - if you want to take in the museums, you would need to make Florence your base, but if you just want to walk around and enjoy the cityscape, it's sufficiently compact that you could do that in a day. That said, Lucca is quite lovely, and I love the idea of staying there that long.

Train to Pisa, change to a train to Lucca. Basically one train every hour. Takes 2 1/2 to 4 hours depending on the type of train available to Pisa. The fast trains cost more, the slow ones are cheaper but make lots of stops on the way.

If you have a desire to spend a reasonable amount of time in Florence visiting museums then it serves no purpose spending all of your time in Lucca - spend 2 nights in Florence on the way. From Lucca you can spend a day visiting the Cinque Terre as well.

How to Look Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it. This doesn't sell tickets for Italy, but gives you pretty user-friendly schedule examples for the current fall period. Lucca and Pisa are also connected by a direct bus (less frequent on Sundays). Rick's Italy guidebook covers all of your destinations mentioned, except just transport tips (not sightseeing) for Livorno.

There should be shuttles that take you from your ship to the port entrance. From there take
a taxi to the Civitavecchia train station. This route will take you along the coast to Pisa where you
will change trains to Lucca.