When should we buy rail ticket in advance?

All of your trips can be done with regional trains with similar travel times as more expensive trains. So no reason to buy in advance. Possible exception of Perugia to Fiumicino, which saves about 1 hour travel time over regional trains (3 hours vs 4 hours). But even then, you are only saving about 12 EUR by buying a nonrefundable advance ticket.

All IC (Intercity) and Freccia trains have only reserved seats and the tickets CAN be purchased in advance, you'll be assigned a seat and a car number. It doesn't mean you must buy in advance: if you want, you can. The tickets of IC and Freccia trains going to/coming from the small seaside villages in summer SHOULD be purchased in advance, to be sure you'll have a seat on an airconditioned train departing right at the time you want.

There are no reserved seats on Regionale trains (R or RV on trenitalia.com), so these local trains CAN NOT sell out. For the same reason, getting the tickets months in advance or 16 minutes before the departure time makes no difference.

I think many of your trips will be on Regionale trains, but you can check it on the English version of trenitalia.com. Pick an early time of travel when day-tripping.

Thanks for the good advice.
In purchasing tickets, I assume that ticket agents will be at rail stations.
Last time I tried to by tickets at a ticket machine in Barcelona with my US credit card, I couldn't get it to work.

Both Trenitalia and ItaloTreno have easy to use apps

If you already know what time & date you want to travel, you can buy them on-line and have on your phone or print out the tickets at home. I went to similar locations in June, and I just buy them on-line at www.Trenitalia.com

If you do buy them at the stations, be sure you validate the tickets given to you (insert ticket into machine to time stamp it), or you will have to pay 50 Euro fines on the train. Almost all of my trains had conductors verifying people’s tickets.

I am not sure about having the tickets on my iPhone. My service doesn't work in Europe, unless I have access to wifi.

But tickets bought on line now will be prevalidated, so if you change your mind on where and when you want to go, you will be out of luck. Using the app is the best solution. Then you can buy just before needed.

Sam,
If I can only use the phone when I have wifi, I suppose that I would could do that at my hotel prior to going out to travel?

You should be able to take a screenshot. Just make sure the name, date and any barcode are in the shot. You shouldn’t need wifi to access a photo.