Long distance trains in Italy need a seat reservation. Intercity trains 3€ and high speed trains 10€.
FR is a Frecciarosa ("red arrow”), the highest class of high speed trains of Trenitalia.
Depending on the length of the journey, there can be reservation-free alternatives using regional trains, some of which also cover quite long distances. Check the Trenitalia website and select "regional” instead of the default "main solutions”. Be aware though that the regional trains for the summer timetable (starting 12 June) have not been published yet.
You can book reservations in advance. Usually a few days is enough or even on the day itself (also at the ticket office), although there's always a risk that a train is full. The best way to book online is via ÖBB using these steps.
Athens - Thessaloniki can easily be done in one day. Check the timetable on the Greek railways website. You will also need a reservation there, which is free of charge, but only available from the ticket office in Greece.