Hi there, both are relatively simply journeys using the train services, just make sure that you don't leave it too late to get back to the ship.
For Civitavecchia to Rome the station is only a couple of hundreds yards from the dock gate - just make sure you get the first shuttle bus & early train. When arriving in Rome, there are 5 stations, the first St. Pietro is the closest to the Vatican, the last Termini is the best place to pick up the hop-on hop-off bus service - they have excellent headset commentaries.
At Livorno, you have to take a public bus from the main square behind the shuttle bus drop off point. Make sure you buy your ticket before you board the bus, as it is MUCH more expensive to buy it once on the bus, they also fine heavily if you do not use the franking machine on the bus to validate it. The train passes through Pisa en route to Florence/Firenze. It is possible to see both if you really want to, but the day will be tiring.
The rail website has changed name recently from www.trenitalia.com
This is all in English, and quite easy to follow to plan your days.
Hope this is of use, have a lovely time,
Alan & Katrina