How far in advance should I book train tickets in Italy?

For high-speed, two to four weeks ahead — that's where the cheap fares are, sometimes a third of the walk-up price. Tickets typically go on sale about four months out. For regional trains, don't bother: the fare is fixed, they don't sell out in the usual sense, and buying on the day keeps you flexible. The trade-off with cheap advance high-speed fares is that they're non-refundable and tied to that exact train, so if your day is loose, pay a little more for the flexible fare instead of gambling on making a connection.