In Italy the main train companies are two: Italo and Trenitalia.
Trenitalia covers quite 100% of italian stations and have different kind of trains.
regionale and intercity are the two less expensive that covers both long legs and short distances (about long legs: are cheaper but takes like forever because stops quite in every station, for istance venice-rome with regionale it's about 8 hours and 4 changes).
freccie are the trains for long legs, less stops, faster and more confortable. There are three types of frecce: argento, bianca and rossa: here a detailed description from trainline site (in fact it's a translation of the italian version you can find here from trenitalia site which is in italian, sorry i can't find the correspoding page in english in the official site...). In this trains you will find the better quality, and in freccia rossa the best (like the exclusive in your picture).
About italo, it covers the main legs (it's expanding it's offer) and have basically 4 classes: smart, confort, prima and club exclusive here a description from italo site
You can find other companies, but they cover small legs and only regional, the quality of service is usually similar of trenitalia regionale/intercity because are used for short times/distances and are usually "specialized" in particular routes (for example trenord for lombardia region and milan malpensa airport).
Edit: Be aware that if you need to go, for example, venice-rome you can buy venice-milan from italo and milan-rome from trenitalia but since are different companyes if the first train (for some reason) it's late, the second won't wait for you (and if you don't have a tiket that let's you to change day/time you also will loose the money)...if you buy all the leg in one single tiket from the same company, in case of delay or the second train will wait for you, or you will be able to catch the train after with the same tiket (in this case is better to explain to the conductor that you missed the previous because of a delay showing the tiket as soon as you take the train) - thay are oblied to give you a solution if the missed coincidence it's caused by them if you buyed a single tiket for the leg.