My SO and I went on a Mediterranean cruise back in 2009. The cruise itself started in Venice, went to Croatia, 4-5 Greek islands, Turkey, Naples and ended in Civitavecchia. We spent approx. 9 days before the cruise on our own, flew into Rome (spent 2-3 days), took a train to Florence (spent 4 days) and took another train to Venice (where we spent 2 days before boarding the ship). I have to say that seeing all of the Greek islands and other port cities was fantastic. You'll absolutely love it.
I second the recommendation to check out Rick Steves' books. They are generally great at giving overviews and suggestions on what is doable in a given time frame (he's all about efficiency). If you aren't familiar with his books, you can grab most of his books from the local library to see if they are to your taste (although, they may not be the latest edition). If you like his style, you can always purchase the latest edition from your favorite retailer.
For Livorno/Pisa/Florence, it depends on whether you plan on returning to Florence in the future. Florence probably requires at least 2-3 days to fully enjoy (the museums, the Duomo, Ponte Vecchio, the local ambiance). According to your itinerary, you have 12 hours at Livorno. I believe it may take you approx. 1.5-2 hours each way to go from Livorno to Florence (total of 3-4 hours roundtrip), which leaves approx. 7-8 hours (very rarely do you get off the ship right after it docks and you usually need to return to the ship at least an hour before departure) to see Florence itself, which isn't a whole lot. Surely you can do a few of the major sites in Florence in that time frame, but it would be very rushed. As an alternative to Florence, Pisa and Lucca are both quite close to Livorno and neither require a large amount of time to visit as we saw both in one day (as a day trip from Florence). As you probably already know, Pisa is known for the Leaning Tower (it's shorter than I expected it to be) and Lucca is a small, charming walled city. In Lucca, we rented bikes and cycled on the wall around the entire town. Rick Steves' has a lot of detail on Florence as well as Pisa and Lucca.
For Naples, we did both Pompeii and a private driving tour along the Amalfi Coast in one day. IMO, the Amalfi Coast is absolutely picturesque and I would highly recommend it. For the Amalfi Coast tour, we actually stopped in the town of Amalfi and had time to walk around and take pictures. We had lunch at a restaurant in the hills before spending the afternoon in Pompeii. The entire day spanned from 8 AM to 5 PM.
General recommendation for Italy: I would highly recommend grabbing gelato anywhere you can in Italy. We ate it everyday (sometimes twice a day!) because it was so delicious.
I've done a lot of cruising (11 cruises, all with Princess) and we definitely avoid any official/ship-offered excursions. They are always overpriced and you can always find a better vendor for cheaper. In some ports, we just explore the town on our own using a Rick Steves' book as our guide.
Cruisecritic.com is a great resource for specific ports, but it may take some time to sort through all of the information (just like any other forum, i.e. FT). If you want to save yourself some time planning, you can just grab a Rick Steves' book.
