Buy trenitalia tix to Rome at the station or on the app?

It depends on whether you are in charge of when you arrive at the station. If you are willing to commit to a time and train fast train tickets can be bought ahead at a significant discount. However, if you are getting off a flight or other activity and can't guarantee when you'll arrive then just wait and buy them when you know you're ready. There are trains essentially every half hour between the the two fast train companies.

I recommend getting the Trenitalia app for checking schedules, buying tickets and checking on trains.

Have a great trip,
=Tod

Using the app to buy the ticket when you arrive at the station is also an option... Ticket vending machines sometimes have queues.

What you can do if you are not sure which train you will take is buy a "BASE" ticket for a train you are pretty sure you will be able to take. If you end up earlier at the station you just get out your phone and move your reservation to the next available train. This is what I do on business trips when I do not know in advance when my business will be finished.

I'd buy on app and as mentioned get the base fare if you may need to change. Usually you can do the check in also on the app which for longer high speed train rides I like so that I am not bothered when they want to check the ticket.

If you've not yet done so, the app is definitely the way to go. My wife and I use it with confidence owing to it's offering the discount for online tickets and the ease of having the ticket on our phones. It's one less piece of paper to have to keep track of...very convenient.