We spent a week in Rome prior to our cruise. It is a WONDERFUL city! We felt the time we spent there was a perfect amount of time to see everything, but we could have done it in three full days. We did private tours of ancient Rome, the Jewish Ghetto and the Vatican. We did a small group tour of Ostia Antica. I am happy to provide tour information and details, and will probably post more on those things on Trip Advisor.
Sorrento is a great "jumping off point" for other towns on the Amalfi Coast. It has a nice little shopping area where we picked up lots of gifts for those back home. Sorrento has AMAZING citrus fruits and we brought back lots of lemon soap, liquor, and lemon olive oil! The transportation Azamara provided was excellent. It is a quick and easy walk to the train station to catch a train to Pompeii (which we did), Herculaneum, Naples (for the museum), etc. There are some great tours to Amalfi and Positano as well.
We took a private tour of Malta. The all-day tour took us to every nook and cranny of the island! Some other guests used the "Hop On Hop Off" tour bus and said it was great. Another couple we talked to used the public bus and said it wasn't the best way to see the island because the wait for the buses were up to an hour each time! Malta was a lovely surprise. The island is full of sites to see and things to do. We really enjoyed Mdina and Rabat, and even had time to go to the beach! A very nice day.