If your cruise ship docks at one of the industrial wharfs in Livorno Port (which all the big ships do), there will be a cruise shuttle bus service into town which drops off and picks up in Via Cogorano, alongside Piazza del Municipio, opposite Livorno town hall (Comune), just behind Piazza Grande. There is usually a charge for this. In 2016 it cost €5 per person, round trip.
There is a Tourist Information Office in the building that separates the Piazza del Municipio from Piazza Grande. Here you can get maps and other information about Livorno, as well as tickets for the boat trip along the canals. Some of the smaller cruise ships dock in the old Medici Port (Porto Mediceo) which is within easy walking distance of the town centre of Livorno.
To find out in advance what there is to do and see in Livorno, click here
If you are looking for information about other cruise ports, visit Tom's Port Guides for a wealth of accurate and practical information.