Hi,
I am sorry for the delay, but my info posted yesterday evening has gone missing! So here goes for another try.
You will almost definitely be taken by the ship's shuttle bus to what you refer to as downtown Livorno. Its sort of a side street with access to quite a large square where you are dropped off. In this square you will see a number of bus stops. Look for the info that says No 1 as this is the bus you want. Buy your ticket from the newsagent's kiosk immediately to your right. (1Euro) This bus only goes as far as the railway station so you cannot get lost. Buy your ticket to Pisa, Florence or wherever you would like to go. If you want to go to the 'Field of Miracles' (Leaning Tower) it takes approximately 15 minutes. Don't forget to validate the train ticket in the yellow boxes situated on and around the platforms both on the outgoing and incoming journeys. If the inspector comes round and it is not date stamped, you will be fined on the spot and he takes no excuses i.e. language problems, forgot etc., You will need to take a bus for the very short ride to the Leaning Tower site and this will be waiting outside of the railway station just across the road.
Livorno to Florence. Florence is lovely but in my oponion warrants more than a day. There is just so much to see. The journey time is 1hr 22 minutes and the cost is approx Euro9. If you do decide to go to Florence be prepared for a long day and make an early start. If you want to check the train times look on the trenitalia web site. I think I lost my post because i gave the full web addres of this site. It starts with www and ends in com! If I can help any further just let me know. Enjoy the med.

Just a little word of warning if you are going to Florence. Some of the trains require a transfer at Pisa, and others go on to Florence.

If you are going to Pisa to see the Tower etc. you can walk from the station. It will take about 20-30 minutes depending on walking pace, but lets you see the town. Basically you go straight out the main entrance of the station and keep walking straight onish. You go through the town and over the river, on for a bit and ultimately bear left. The Tower is well signposted. You can still take the bus back to station, and I would not recommend walking back if you have not walked there and memorised the route.

A good tip these days is to uses somewhere like http://www.maps.google.co.uk.

Zoom in on the area you are planning to walk throught/around and print of a street map AND the satellite map which will help with land marks.

eg for Pisa : tower at the top station at the bottom
http://maps.google.co.uk/maps?f=q&hl=en&geocode=&q=pisa+train+station&ie=UTF8&ll=43.715442,10.401564&spn=0.014486,0.026736&z=15

Might need to be zoomed in and a couple of maps printed to get enough resolution depending on how good you are with directions.