brigitte,
I second the suggestion from Kathy to buy a good guidebook and pack it along on your trip. The Rick Steves books are excellent, and provide a wealth of information that will help your trip to go smoothly. Most of the books are also available as E-books, should you be travelling with a Kindle or iPad.
Be sure to pack Money Belts, and if travelling with electronic gadgets, you'll need to check each one to ensure it will operate on 220 VAC. You will of course need Plug Adapters.
Regarding the trains, you may find THIS short video helpful. It's a bit dated but still good information. To provide a summary, the express trains (Freccia, Intercity) have compulsory seat reservations. The tickets for those are specific to train, date and departure time, and can only be used on the one train listed on the ticket. If you board the wrong train by mistake, you'll be subject to hefty fines which will be collected on the spot!
On the high speed routes, you can also use the competing Italo Treno trains. If you wish to buy advance tickets for the high speed trains, you might try www.trainline.eu which is very user friendly. They sell tickets for several rail networks in Italy (and other countries) at the same prices as the rail networks.
If using the slower Regionale trains, Metro, Buses, etc., locally purchased tickets must be validated (time & date stamped) prior to boarding the vehicle on the day of travel. Failure to validate may also result in hefty fines. The validation machines are easy to spot as they're usually located close to the tracks - https://goo.gl/images/BbPNZM . On buses the machines may be onboard the bus.
Vaporetto tickets in Venice must also be validated, but this is electronic. Touch your ticket to the reader and if you get a green light, you're good to go - http://actv.avmspa.it/en/content/validating-actv-travel-documents-simple-0 .