Should we buy our tickets before we leave for Italy, or buy them as we need them at the train station? We are going from *Siena to Cinque Terre; *Cinque Terre to Rome. Is there ever any problems with trains being full and not being able to purchase tickets/
Advance purchase of ticket for Italy is unnecessary and more expensive.
Regional trains do not have reserved seats and there are no limits on the number of tickets they sell for those trains (in fact, most of those are open tickets, valid on any train you want, so any regional train can be full, whether you bought tickets before you left or not.
Faster trains have reserved seats, but few would be sold out a day in advance. Just get your tickets at the station. If you are really concerned, get it the day before.