You can use the high speed trains of Trenitalia (Frecciarossa and Frecciargento), they run more often then the competitor italo. You only need to book an additional reservation for 10€. 

For regional trains you can use almost all trains, but there are some exceptions. If you can give us some more Information where you would go and the date, we can help you. 

I think you may have misread. If you ONLY want to use the local/regionale treni, then TrIt offers much cheaper passes for only these treni from their own. And some of the regione In IT also offer passes only valid-but for all transportation-in their area. F.e. LO-Viaggio in Lombardia (around Milano/Brescia and the lakes there) or for Umbria.

The nasty thing is that in some of the provinces Tr-it has merged with the local treni and got a new name-it is still unclear if these all accept the passes

The tipical 1st time visitor to IT mostly wants to visit the most famous cities and between these most will use those superfast Frecce treni. The locals/regionale are mostly for those with enough time or wanting to visit specific places-or maybe their famiglia!