trains

Italiarail is not a railway company. They are a reseller of train tickets.
The two railway companies running high-speed trains between the major cities are Trenitalia and Italo. Buy your tickets with them or with an agent which sells them at the same price, such as Trainline or Loco2.

Ton has given great advice.

And where possible, take the freccia trains with Trenitalia or any train with Italo. They may cost a little more, but you get a reserved seat and a fast and comfortable trip with fewer stops than a regionale. The slower trains are false economy, especially if time is precious.

But if you trip requires some time on a regional train (because the origin or the destination is a smaller place that doesn't have Freccia service), you may pay more but not save any significant time by choosing a Freccia + regionale option rather than an all-regionale route. This will be obvious when you look at the schedule. What can happen is the Freccia zooms right along, then you spend a lot of time cooling your heels at the point where you change to the regionale.