The most economical train option is the Eurostar, abbreviated ES*. This is the most modern train, and the fastest. Eurostar trains connect popular tourist destinations and major Italian cities, and feature air-conditioned compartments. First-class service and second-class service are available. Seat reservations are automatically assigned for Eurostar trains and tickets may be booked online in advance or at the stations.
Also connecting major cities throughout Italy as well as smaller cities Eurostar doesn't service are the two city train lines, Intercity (IC) and Eurostar City (ES*City). These are the most commonly used trains and are the most practical means of cost-effective and time-effective train travel. There are newly updated Intercity Plus trains that are more comfortable than their older counterparts, but both offer first and second-class service. Air-conditioning is available, but may not work as consistently on the older Intercity trains. Reservations are not required, though the seats are still numbered in the compartments. For a longer route it's still a good idea to make reservations.
The Treni regionali (R), diretti and interregionali are the slower regional trains that offer the cheapest means of travel around Italy. These are the "local trains" that run more closely with school and work schedules. The latter two are the regional trains offering longer routes but all three serve the smaller towns and provincial areas. Though these are the least expensive trains, they are also the least reliable for time efficiency. If you're looking for the leisurely train ride rather than a tightly scheduled route, this is a good bet. The regional trains only offer second-class seating without seat reservations.
If you're traveling overnight, consider sleeper trains, which also vary in price and comfort, but tend to have at least Intercity standards for their one to six person berths. This may allow you to reach some destinations at better cost because you're combining "hotel and transportation" into one price.
Also check for any extra booking or tariff fees that you need to pay when you make your train reservations or validate your tickets at the station. These are minimal, but could add a surprise to your planned budget if you don't know to look for them in advance.