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In Livorno take the bus to the port gates, buy a bus ticket for the #1 bus for €1 and catch this to the station. Trains to Pisa take around 15mins.
Pisa can be visited very cheaply by using the shuttle, bus and train. Florence is more difficult by train and the ship has been missed by some who messed it up, the ship normally do a shuttle run to Florence which is reasonably cheap and I, personally, wouldn't try to combine the two cities.
Rome is easy: Take the shuttle to the dock gates (free), walk to the station then catch the train to Rome Terminus Station. If you search the "Rome" section in the subject section above you will find a tour of Rome I did from this station, the ticket which must be validated on the platform costs about €9 and includes Metro and buses. If you just want to see Vatican City get off one stop earlier and get out at St Petro Station, .... Neil