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If you type 'Livorno' into the search box you will find lots of threads on this subject. Been covered many times before - as with Pompeii. Livorno is a huge industrial port - To get to the train station you need to take the cruise shuttle to downtown then the No1 bus to the station: buy bus ticket from news-stand (round trip) where you are dropped off. Pisa is actually rather dull, and the leaning tower takes a few minutes to see. Personally I recommend visiting the hidden gem of Lucca. Trains from Livorno to Pisa take 15 mins, then change there for the train to Lucca. On the way back give yourself 90 mins to walk to the leaning tower to take the obligatory picture and then catch the next train back to the ship. Taxis are loath to take people the short hop from pier to station as they would rather be hired for the whole day to drive up to Florence etc.
Thank you for all your help. (Unfortunately I couldn't get the "search" to work when I tried yesterday but will try again!) .
Train times are easily found on http://www.trenitalia.com/en/index.html.
Naples: When you exit the ship walk across the street to the newstand. Buy a bus ticket to the train station. Walk back across the street and take the #1 bus to the train station. Don't forget to validate your ticket on the bus. The bus ride was is about 10 mins. Take the Sorrento train and get off at Pompeii Scarvi. The entrance to Pompeii is right behind the train station. it takes about 15 minutes on the train. There are private tour guides waiting to be hired, or go to the information booth and get the free map. You will not find the trains listed on the Trenitalia or most other train web sites. The train line between Naples and Sorrento (also goes to Pompeii) is operated by a private company (not the Italian Trenitalia company) called the Circumvesuviania Line. The trains do depart from the lower level of the main train station just like the other train line, but its schedule is often omitted from many web sites. There is generally a train every half hour between Naples and Sorrento. The train costs €3.20pp from Naples to Pompeii and entrance to Pompeii is €11pp. One person should get the audio guide for €6 and they can summarise the key facts for others.
And finally, I recommend that you join the meet&mingle for your cruise - RCI throw you a party!