Day trips-buy in USA or wait until in Italy

Forgot to add day trip from Sorrento to Capri to prior list

There is no blanket answer. It depends -------- Without checking each route, my guess is most of your travel will be via Regionale train. There is no advantage to purchasing Regionale train tickets on-line. Same price on-line or at the station/ticket machine. So buy at the station and validate prior to boarding. Regionale trains have no seat reservations. Just get on and go. No seat. Stand till one becomes available. All other trains in Italy require a seat reservation and it comes with the ticket. If it is a faster train then there can be discount tickets available but they come with restrictions -- no refund, no change. Check your schedule at trenitalia.com for type of train and prices.

What Frank said.
If you want to avoid the whole standing in line at a kiosk to buy a paper ticket and then having to validate it before boarding routine you can download the trenitalia app onto your smart phone. Your electronic ticket(s) are sent to you as an attachment to a confirmatory email. Search and purchase at your leisure. No validation required - No hassle - works great.

I tend to take short hops on trains, so most of my Italian tickets were on regionale and I just bought them once I got to Italy. It is the case, however, that ticket lines at Italian rail stations can move very, very slowly. I assume part of the problem is that a lot of the people at the staffed counters are making tricky purchases (international tickets for a family, for example). Using a vending machine is a good idea. If for some reason you cannot do that, allow plenty of time to buy your ticket; otherwise, your train may depart while you're still in the ticket line.

You can check to see whether any of your trips will offer savings for purchase in advance by going to trenitalia.com and looking up each one. You'll need to use the Italian city/station names:

Roma (Tutte Le Stazioni), Firenze (Tutte Le Stazioni), Milano (Tutte Le Stazioni), Venezia (Tutte Le Stazioni), Padova and Napoli (Tutte Le Stazioni). If your Lake Como destination is Varenna, the station will be Varenna-Esino. Be sure to use the correct day of the week when you look up the schedule. Any train called "Frecciaxxxxxx" requires reservations (as do EuroCity trains, and there may be a few others), and the fare will tend to increase as you get closer to the travel date. From Venice to Padova you will have a choice between Frecce and regionale. The Frecce will generally save you only 12 minutes but will cost about 15 euros more if you buy the ticket at the last minute.

Edited to add: As Frank points out below, my wording was careless. What I meant was that if you wait to buy tickets on one of the fast trains, you will likely pay more. Other passengers may have long-since snapped up the limited number of cheap tickets that were initially available.

..... fare will tend to increase as you get closer to the travel date..... I am sorry but that is just incorrectly stated. Fares are fixed, do not change. You are not dealing with a US airline. You have three levels of fares -- Super Econ, Economy, and Base. The SE and E fares are discount fares, limited number available per train, and once sold out, they are gone. The Base fare is the standard price and is the same from 120 days out to the day of travel. What happens is as you get closer to the day of travel the discount fares will be sold out earlier leaving the Base fare as the only available fare but it never changes in price.

The other thing that gets overstated besides pickpockets is lines at ticket counters. Generally most lines especially for ticket machines are short -- maybe 5 mins. Often no lines in the larger stations. Same for the ticket windows. There can be lines at peak travel times or holidays -- not denying that. But generally lines are reasonable. One way to avoid the last minute rush is to buy the ticket the day before or if arriving via train buy when you first arrive. That way if you find a long line, then it is inconvenient as opposed to potential missing your train because you are standing in line.

There is a recent change (I believe, no current experience) for purchasing Regional tickets. It use to be that a Regional ticket could be used anytime within a 60 day window -- or something like that. Now I think you have to specify the date of travel when you purchase a Regional ticket and it only good for travel on that day. I could be wrong but that is my current understanding and if I am corrected I will remove this.

And finally, since you obviously are new to train travel be careful with chosing "Tutte Le Stazioni" That is not station anywhere in Italy. It simply means "All stations in _____." Most cities have more than one train station so you need to know the station so you don't accidentally get off in the suburbs. Just google the cities to find the main train station.

I believe the only trip that might possibily be on a Freccia train is Venice to Padua. However, the less frequent regional trains on that route are just as fast at a fraction of the price. I don't recall there being any advance purchase discounts for the boats from Sorrento to Capri, and bus tickets from Florence to Siena aren't discounted. For your trip to Burano, it might make sense to buy a 1-day Vaporetto pass if you don't buy a pass for your entire stay (the multiple day pass can be a good value).

To follow Frank's comment as an example --

The Regionale train to Padva is 49 mins/ 4.35 Euro

The fast train is 26 mins with a fare range of 14.90 (SE ticket) to 19 Euro for the Base fare. You pay 15 euro to save 23 minutes.

For most of the trips you've mentioned, the trips will be via Regionale train so there won't be any discounts for buying advance tickets. Just buy tickets locally either using a Kiosk or at a staffed ticket office, but as Frank mentioned be sure to validate prior to use on the day of travel or you'll risk hefty fines.

Your trips....

• Milan to Lake Como (using Varenna as an example) - Regionale operated by TreNord - trip is 1H:03M each way. It's a good idea to buy tickets in Milan for both the trip to Varenna and return.
• Venice to Padova - you can use either Regionale or express trains to Padova but not much reason to pay the higher price for the fast trains as the travel time is about the same. Trip to Burano will be via boat so just buy locally (those tickets will also have to be validated).
• Florence to Pisa and Siena - many of the trains to Pisa will be Regionale; the trip to Siena is easier by Corse Rapide Bus as you'll be dropped in the centre of town rather than at the bottom of the hill where the rail station is located. The bus station in Florence is very close to Firenze SMN rail station. Don't forget to validate your tickets.
• Rome to Assisi - many of the trains will be Regionale
• Sorrento to Naples - you'll be using either the Circumvesuviana or the Campania Express (depending on what time of year your trip is taking place). I don't think it's possible to buy advance Circumvesuviana tickets nor would there be any discount for doing so.

If you don't already have a copy, I'd suggest having a copy of the Rick Steves Italy guidebook at hand as you plan your trip.

Thank you all for your insight. Evidently, Italy is a bit different than Greece. The last 2 years we visited Greece and found that we paid much more for bookings on line from the USA than getting tickets locally.

We do a lot of regional trains. Using a kiosk is easy. Start by changing language to. English. If you buy tickets for the day of travel they are good for 4 hours from validation. (Don’t forget to validate before getting on train.) Or you can select a future date a couple days out, and valid for that day. If it’s a day trip you can choose round trip, and you’ll receive two tickets. Wait to validate the return ticket until you are about to board on way back. Most kiosks accept credit cards and cash.

When you select the train you want, be sure to choose the regional if that’s what you want (big letter R)
I generally do screen shots of the train options so we know what train number and departure time and how often trains run. (Good for finding your train on departure board)

Lines at ticket windows like Florence, Milan, and Rome can be long, so yes kiosks. There are often trenitalia personal to assist if you gave questions.

Even in places of modest size, the staffed counters can be an issue. It's not that the line will necessarily be long, but that it may move at a snail's pace. I suspect the computerized ticketing system maybe be somewhat inefficient. Or it may be that most of the people in line have complex transactions to conduct or do not have Internet access, so the ticket-seller has to begin by explaining all the options available and their cost, then the customer has to choose one.

This is a problem I haven't encountered outside Italy except at the nightmarish Atocha Station in Madrid, where the delays are grotesque (waits of multiple hours). A ticket machine is always a better option if you know what you want and don't run into a software, hardware or payment issue.

How do you validate your ticket before getting on the train?

Look for those little green or yellow machines, they are all over the train platforms https://www.seat61.com/Italy-trains.htm#Travel%20tips

You don't need to validate train tickets if they are for a specific train/date/time. The purpose behind validation is to prevent the ticket from being used again. Not an issue if it can only be used once.