The best way to book seats in Italian day trains is via ÖBB using these steps. That way, you avoid Interrail's booking fees. Of course, you can also book at an Italian ticket office.

Currently, you can book until 11 June. Trains from 12 June are expected soon - Trenitalia have already published the summer timetable on their own website, but it needs some more time to be distributed.

Passholder seats in Italian domestic trains are not quota-controlled, so there's no need to book very early. Also, keep in kind that there may be reservation-free alternatives with regional trains, but the summer timetable for the regional trains has not been published yet.