Livorno railway stationThe Train Service from Livorno

From Livorno there are regular trains running south serving the coastal towns, Rome and beyond, north towards Pisa, Genoa, Turin and beyond, and east towards Florence, via Pisa. Tickets must be stamped before beginning your journey at one of the yellow machines in the station or on the platform.
Getting to Florence by Train
Here is some info about the trains to Florence from Livorno. Doing it by yourself is usually cheaper than an organised tour, and you will have a few hours in Florence, but if you are on a cruise make sure you get back in time for the ship's departure! 
If your cruise ship docks in the industrial port, as most of the big ships do, there will be a shuttle service which takes you into the centre of town. From Piazza Grande, which is the square right behind the square where the shuttle bus stops, you can catch a no. 1 (most direct - about 10-15 minute ride) or no. 2 bus to the station. Buy a ticket at one of the news kiosks and stamp it in one of the yellow machines when you get on board. The tickets cost €1. You may as well buy a couple each in the morning so you are equipped for the journey back.
Alternatively, you could get a taxi from Piazza Grande (or maybe even from the ship - see below).

The trains to Florence run about every hour - 08.11, 09.11, etc. and it takes about 1 and a half hours. On Sundays there is no 10.11 train, so you should aim to get one of the earlier ones (there is a train at 10.00 but you would have to change in Pisa, getting into Florence at 12.00). Even if your ship docks early, it would be difficult to get the 8.11 train as I don't think the shuttle buses begin that early (but check with the ship).
There are usually taxis waiting at the ship, but they are not usually very keen to take a local fare as they are trying for full day further afield work, like Florence and Pisa. However if you're there early they might make an exception.
The second class train fare is about 13 euros return. Again, make sure you stamp your tickets BEFORE you get on the train, and before you make the return journey too. Small yellow machines on the walls all over the stations.

RETURN: There are trains from Florence at 15.27 (arriving Livorno 16.49), and 15.57 (changing in Pisa, arrive Livorno 17.06). Most ships leave around 6pm, so the 15.27 is probably the latest you can leave it (but please check with the ship about latest arrival times for passengers). On the way back you could save time by getting a taxi from the station directly to the ship (make a note of the dock name - probably Varco Valessini or Varco Galvani, but there are a couple of new ones too).

I don't think you will need to book in advance, unless you want to gain some time. If there are lots of passengers from your ship doing Florence by train, there might be a queue at the ticket office.