Unless things have changed, there are schedule releases for tickets in the beginning of June-ish and December-ish. Some routes can have trains listed before then, I can't tell which ones why. If you buy them as SOON as they go on sale, you can save a BUNCH. As the cheaper fares get sold, the fares go up until the "day of purchase" price that you are doing when you arrive at the airport station. I was traveling from Lake Garda to Florence once on a Saturday, and when the tickets got released, there was a promotion for 2 for 1 first class seats on Saturdays, so we even saved money to take that trip! It would have been sold out if I would have waited.
You can use the app or the website to purchase in advance (I rather do the pre-planning on my computer, it's easier because you can see more and compare options with ease), but if you know when you will be traveling, get them asap. The cheapest fares are usually the most restrictive regarding changes and usually not refundable, so if you miss, you are out the money. I always aim to get to the train station 30 min ahead, so if any transit delays happen I have a buffer.
I also travel between check out/check in times to utilize my time well, like eating at my favorite restaurant near the train station in Florence at noon, get on a 1:30 train to Rome, and then arrive at my lodging in Rome 3-3:30ish.