Sara, a couple of basics for Italian trains. First, when you buy a ticket from one city to another and there is a train change along the way, you will receive one ticket for each train that you are on. Second, there are different types of train categories in the Italian train system. If you ride on the Regionale (R) categories, which is the lowest category, you will receive an open ticket good for 60 days. No date, no time and no seat assignment. The ticket will have the departure and arrival locations on the ticket along with the class (1st or 2nd) and the number of passengers. You can take any R train making that particular run anytime during the 60 days. Also, you can get off the train anywhere along the way, stay for as long as you want and get back on a later train making the same journey. An example of this type of ticket is from Florence to La Spezia. If you have a R ticket, you can get off at Pisa, visit the leaning tower, return to the train station hours later and hop back on a later train to La Spezia. Reservations or seat assignments are not permitted on the R trains.
The next category up is the intercity (IC) and ICPlus trains. These are much like the R trains except that you can pay about 3 Euro for a guaranteed seat assignment.
Next up are the high-speed trains. These are the Italian Eurostar (ES), Alta Velocita (AV), Cisalpino (CIS), EuroCity (EC) and night trains. These trains will have the date, time, train number and seat assignments on the ticket. No fooling around here. If you get off one of these trains midway in transit and your train departs, the remainder of your ticket is toast. (See my next message)