Buying train tickets within Italy

Can I buy train tickets online for train travel within Italy

Yes, although it isn't necessary.

what's the best website to use?

Trenitalia is Italy's national rail system. Italo is a private line with competitive rates but mostly services larger cities. There are smaller regional, commuter lines as well - like the Circumvesuviana in Campania - so it depends on where you're going. Some posters like the Trainline site, which is a 3rd party booker but they find it easier to use.

http://www.trenitalia.com/tcom-en.
You need to use the Italian names for the cities when you input your departure/arrival destinations

http://www.italotreno.it/en

https://www.thetrainline.com

Is it advantageous to buy them in advance, even for shorter trips?
Florence to Bologna for example

It depends on the type of trains which run between two destinations. If they're a faster train with seat reservations, then it can be worth booking in advance to take advantage of any (non-refundable, non-changeable) super economy fares which may be available. If they're slower regionale trains with no reservations, then just buying at the station the day of the journey is a better way to go.

I just did a test run of trains between Firenze S.M. Novella and Bologna Centrale and the best choices are faster trains with seat reservations: only 35 minutes or so, and some super-ecomomy tickets might be available depending on your date and desired departure time. The Intercity train for this route takes over twice as long to get there, doesn't have many runs, and isn't less expensive enough to warrant the savings, IMHO.

Do understand that if you purchase super-ecomomy tickets then you are committing to a certain train at a certain time. Miss it and you have to buy another ticket that'll likely be base (full) price. Economy tickets can be changed once but only by paying the difference between purchase and full price. Full-fare tickets can be changed for no cost.

If you want to learn more about train travel in Italy, this is the definitive guide:

https://www.seat61.com/Italy-trains.htm

When I purchase tickets, the site that I have used is Trenitalia. On shorter routes where you use Regionale trains, there's no need to purchase in advance as there are no assigned seats or cost savings to do so. If you are taking high speed trains, it's a different story. These are all assigned seat tickets and the closer they are purchased to your travel date, the more expensive they become. The only caution in purchasing them is that they are not flexible. If your plans change or if you miss your train, the tickets are worthless; they can neither be exchanged nor refunded.

When searching for tickets on the Trenitalia website, you must use Italian city names: Firenze, Roma, Venezia. Also make sure that you choose "All Trains" so you'll see Regionale As well as high speed. On the Florence to Bologna segment, high speed train takes about 35 minutes vs. 1:30 to 2:30 for Regionale. Of course there's a savings on ticket costs. Sometimes the difference in time between the two classes of service is minimal and there's no reason to be locked in to the more expensive option.

We travelled in Italy last September for a month and used only trains for our ground transportation. We did not buy any tickets ahead and just purchased them at the kiosk when we got to the station and never had a problem. We did not want to have a strict schedule as to which train we caught on a certain so the flexibility worked well for us. If you know your schedule and like the certainty of a certain train on a certain day I would purchase ahead of time on the Trenitalia site. "RainPlanner" is a free app that will tell you the train schedule from point to point and the app is usable even if not connected to Wifi. It worked like a charm. Our route was Milan, Varenna, Florence, Vernazzo, Naples, Positano and Rome. If purchasing at the station don't forget to validate your tickets before boarding!!!

If purchasing at the station don't forget to validate your tickets
before boarding!!!

If you have an assigned seat/carriage for a specific train - regardless of how you bought your ticket - you don't have to validate it.

excellent thank you

I was in Italy last week and took trains all over the place. I used the excellent Trenit app from Trenitalia to buy tickets with my phone - nothing to print out, just show the ticket on your phone. I used my Paypal account to make the payment from the app, though I probably could have used my US credit card (which I later used on the Trenitalia website itself to buy a ticket to Nice, which I couldn't buy with the app).

I also used the ticket machines at train stations a few times. I used my chip and PIN credit card there. (You can't purchase in the app within some period, maybe 15 min, before the train departs, and sometimes I cut it closer than that so used the machines.) The ticket machines are easy to use, but I'm not sure how many of them take cash (assuming you don't have a chip and PIN credit card).

Make sure you validate all regional train tickets before getting on the train!!! Even if you just bought the ticket from a machine, stamp it before getting on the train in the validation box.