Buying train tickets

The city and the station should be mentioned, not all trains call at all stations. Stations can be easily found on google maps.

Start typing italian city names in the From/To fields and a drop down menu will display all relevant stations.

"Tutte le stazioni" means "any station". For example, if you are leaving from Rome and you pick "Roma (tutte le stazioni)" from the drop down menu, the site shows trains departing from all stations in Rome.

www.italotreno.it/en is the site of the private competing company that runs only high speed trains on major routes. It works in the same way, but they serve less cities and it may seem easier to use than trenitalia.com/tcom-en

You should get used to italian city names, you won't see any "Florence" or "Rome" on signs, boards and monitors.

I don't believe it's possible to purchase a ticket online without specifying a departure and an arrival city. If there are multiple stations in the city, then I believe you'll also be required to pick a station. The station selection would occur based either on your search criteria or the specific train you select. If you purchase a ticket from Roma Termini to Venezia Santa Lucia, you could get on at Roma Tiburtina and get off at Venezia Mestre. However, if you purchase a ticket from Roma Tiburtina to Venezia Mestre, you aren't entitled to board at Roma Termini or continue all the way to Venezia Santa Lucia. That's why station selection is important.

When one buys the train tickets online at trenitalia should the
stations be mentioned ? Or the ticket is valid for all the stations ?

Yes, as Dario stated, you must specify which station you are departing from and which you wish to arrive at. You must also choose a specific day and time. It would be useful to spend some time reading about Italian train travel on this website:

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

Train travel in Italy isn't difficult but there are some important factors you need to know about, such as the different sorts of trains, which require validating tickets, etc.

Editing to add: I would only pre-purchase tickets for longer journeys involving the faster, more expensive trains. Don't pre-purchase for day trips you intend to take which utilize ONLY the inexpensive regionale trains: those tickets are easily purchased on the day of the trip.

All tickets in Italy except for the Regionale trains are specific to (restrict to) a specific train, at a stated time, at a specific station to station, car number, and seat number. Regionale trains are open seating and only specify the beginning and ending station. If going from A to D on a Regionale train you could get on at B. If the ticket is good for B to D you cannot get on a A. The Regionale train ticket is the ONLY ticket that must be validated (time stamped) prior to board the train. Fail to do it and the fine is huge and immediate. The is no advantage and some disadvantage to purchasing Regionale train tickets on line. I would never purchase Regionale on line.

only specify the beginning and ending station.

And the day of travel.

There is no advantage and some disadvantage to purchasing Regionale train tickets on line.

Well, there is the advantage that a regionale ticket purchased on-line doesn't need to be validated as it is valid for X hours from the departure of the booked train.
Kids combine the advantages of both systems buying regional tickets using trenitalia's app while they are walking towards the track or while they are getting on. I guess that's the future.

I would caution thatmit's risky to walk into a train station in Italy 15-30 minutes before departure time if you want to buy a ticket at a staffed counter. There can be a line that takes quite a while to get through. If you're good with using the ticket-vending machines, you have a bit more flexibility, but you still shouldn't cutmit too close.

I don't know about the app time limit, but trenitalia's website sometimes stops selling tickets several minutes before departure times.