Barca was our debarkation point. It's a good shopping city, with some neat architecture and great food. The port was well run, and efficient.
Florence is really a city you need at least 2 days in, and to take it slowly. Much of the beauty comes from the sheer levels of detail around the churches and city, as well as the integrated history of the Medici family patronage in the city's architecture and layout. I'd recommend spending 1 day with a guide for Santa Croce, Duomo and Uffizi. It may be tiring, but is absolutely worth it to go to the top of the Duomo, but not the belltower, just the main dome. For Uffizi, definitely need tickets online, or you will wait for hours. There is great shopping near the Piazza della Republica, and near the Santa Croce. Santa Croce also has a cool leathermaking school inside. The second day could be dedicated solely to shopping. The tour company that RCI uses makes you waste time at a stupid leathershop hearing their pitch, we could have had an extra 30 minutes had we not dealt with that. We saw our tour guide in Rome the next day with Gucci bags up and down her arm, shopping -- makes you wonder what kind of kickback she was getting.
We had an amazing tour of the Amalfi coast. Sorrento is quaint, with great little shops, and a small town atmosphere. Positano was an interesting stop, very vertical, but maybe not worth it as there is limited shopping, though I did have the best pizza of my life there. Ravello is high above the water, with spectacular vistas, and some decent restaurants. You may not think there is much there, but spend about 90 minutes walking around for some killer photos. Amalfi itself was super touristy, and there wasn't much more to it than that. We skipped Capri, but having been there before it is great. Also skipped Pompeii, but it's neat as well if you have a knowledgeable guide to tell you the history.
The problem with Rome on a cruise, is that it is a city with so much to do, at least a week worth, and you lose 90 minutes in transit each way from Civitavecchia. Plus, we booked a train, and were given a bus somehow instead, which got stuck in traffic. We only had 5-10 minutes at some places, like the Pantheon, and had to entirely skip the Castel San Angelo. It's a long trek from the Coliseum to the Vatican, and you need 3 hours for the whole Vatican at a minimum. So you're literally running past all sorts of cool things. There's really little time for lunch if you don't want to skip a major site like the Pantheon or the Vatican. And you still really need a tour guide for the Vatican, or you'll waste time. Overall, RCI handled this stop rather poorly.
Nice was boring. We stopped at a small flea market (normally a flower market) but there wasn't much going on. We then went to Eze, which is a beautiful medieval mountain town with some 80 residents. It's a rather strenuous climb and was outright dangerous with wet stone -- I'd advise wearing hiking boots at the least, just for Eze, and consider maybe not doing it if you are concerned about falling on hard cobblestone. It's steep, slippery, and no handrails for much of it. I'd spend more time than just an hour -- there's literally a photo vista every five feet, the village is so quaint, and I wish we had more time to look at the shops, or eat at the hotels, some of which looked extremely good (I believe one was even Michelin red-book rated).
Monaco and Monte Carlo are interesting, but I wouldn't spend maybe more than 90 minutes to 2 hours unless you plan on some high end shopping, or lunch at one of the upscale places near the casino. It's really quite neat though if you've never been. Visit the palace, the casino area, and the grand prix circuit. If timing is tight, beware, it can take some time to get in and out during rush hours, and if one of the corniche's is blocked (such as the low corniche when I went) you might face considerable traffic delays.