When looking for property to buy in Livorno you will almost certainly have to deal with a local estate agent, or agenzia immobiliare. If you don't speak Italian I suggest finding someone who does to help you as not many agents speak good English.
The vast majority of property in and around Livorno, as in all Italian towns, is in apartment blocks, or 'condomini'. When looking at property, it is important to take into account the apartment building as a whole, as well as the area in which it is located.
Factors such as the state of repair of a building (don't forget the roof!), the floor the apartment is on, whether or not there is a lift, whether heating is independent or centralised, and what monthly condominium fees amount to, all need to be taken into consideration.
Ground floor apartments can be dark, sometimes have problems with damp, and are more prone to burglaries. Higher floors are fine, if there is a lift.
No lift, however, also generally indicates a lower price.