An excellent way to see Livorno and especially to explore the surrounding area is to rent a car and head for the road! There are several car rental offices in the city, both local and international, and you can of course book a car in advance by telephone, fax or internet.
If you are arriving in Livorno on a cruise, the rental companies will usually deliver the car to the appropriate dock.

Having your own transport means you can visit places that would otherwise be difficult to get to. You could head south along the coast road and drop in on some of the wineries around Bolgheri and Castagneto Carducci, or you could try the hot water springs at Venturina. Closer to Livorno, Castiglioncello is a pretty little seaside resort where you could stop for lunch and a swim (watch out for parking restrictions) and there are plenty of other small bays if it's the beach you're after. Directly south of Livorno, the cliff top road has limited roadside parking with access to various beaches or big stone slabs below - very popular with locals.
If you don't feel up to driving, or you are travelling with a larger group, you can  hire chauffeur driven cars and vans.