Palermo to Reggio Calabria

Hi, Giangilberto from Milan, so my only suggestion is to avoid sites like Trainline or Rome to Rio.
They may be useful for helping you understand the route, but they are not official ticket resellers.
The only site I would rely on is the official Trenitalia site.
From the Trenitalia website, fill in the fields and enter your destination: Reggio Calabria, the city of departure, Palermo, and see the solutions offered.
Usually, for example, there are intercity trains that depart from Palermo and arrive in Messina. In Messina, they are directly boarded on a ferry that takes you across the Strait to Villa San Giovanni.
From Villa San Giovanni, you usually change from an intercity train to a regional train and from there you go to Reggio Calabria.

https://www.trenitalia.com/en.html

Use the Trenitalia website linked by Giangilberto above for trains from Palermo to Messina.
Trenitalia is the official operators of trains. Itarail is just a third party travel agency.

There is no reason to cross the strait from Messina to Villa SG then take another train down south to Reggio.

The fastest way is to take the fast hydrofoil from Messina directly to Reggio. No need to go to Villa SG.

The hydrofoils are operated by Liberty Lines. They have at least one run per hour and the cost is only 4€

The Liberty Lines website is below (don’t see an English page, but you can translate with Google Translate).

https://www.libertylines.it/

https://www.libertylines.it/stretto-di-messina/#

The ticket office is in front of the train station (see below). I don’t think you need to buy tickets in advance.
https://maps.app.goo.gl/xfP9uijZU7p7H8AD8?g_st=ic

The Liberty Lines website is below (don’t see an English page, but you can translate with Google Translate).

FYI
https://www.libertylines.it/en/
https://www.libertylines.it/en/strait-of-messina/

We traveled this route last fall. We took the train Palermo to Messina Centrale, walked to the hydrofoil dock (about 5 blocks) and took the quick hydrofoil to Reggio. We were lucky because we stayed in a hotel in Reggio within walking distance of the dock. We found Reggio Calabria to be an interesting city.