Termini to Salerno - High Speed Rail Tickets

All depends on how confident you are of making the train. If coming straight off an airplane, not recommended because your plane could be late arriving, and if you miss the train shown on the ticket, it is worthless unless it is a full fare ticket. And if it is a full fare ticket, there is not much reason to buy in advance to begin with.
On the other hand, if you are already in Rome and moving on, then you can be 99% certain you can make the train. You can buy them several months in advance on-line and print them out at home and get a pretty good price.

Thanks. Yes, we arrive to Rome on May 9th and will be staying in Positano from the the 12th - 21st. We'll then travel back to Rome and fly back to the US on the 23rd.

What do you mean when you say:
"And if it is a full fare ticket, there is not much reason to buy in advance to begin with."

Ignore that sentence because you will be in Rome already so it does not apply to you. That is only if you are arriving at the airport then buy a ticket for the next train when you get to the airport train station.
I see that on May 12, you can get the 9:53 Frecciarosso 2nd class SuperEconomy for 29.90 EUR per person if you buy it today. If you buy at the station, it is 50 EUR per person. The SuperEconomy ticket is nonrefundable.

I see that too...what is confusing me is the asterick * attached to the fare with a message that states (something like) "this solution is for the following day." What does this mean?

I'm on trenitalia now and I am not seeing it. I clicked on the fare and a pop-up window came up with a box to check "choose seat" which I checked, clicked continue and got a seat map of car 5 showing the available seats I could choose.

Also, I see you can also go for 24.90 EUR pp on Italotreno, with a train at 9:35 from Termini. A lot of people say they are a bit newer and nicer trains than Trenitalia. Easier credit card payment too from what I've heard.
http://www.italotreno.it/en

I'm accessing the site using my phone; will log on using my computer.

What Sam meant is that you can always get the full fare ticket (called BASE price) at the station the day you go up even just minutes before departing. BASE fare is the walk up fare and is always available because trains nearly never fill completely. But if you want supereconomy or economy fares you need to purchase early because those fares sell out quickly. However once you book a specific train at a discounted fare, there is no going back. No changes are allowed. You are stuck with that ticket and if you don't use that specific train the money is gone.

When you search for trains, you have to enter a date and an hour. I usually put in a time that's a couple hours earlier than I plan to leave, to see more options. "this solution. . . " It's possible that you didn't put in the right hour, so as you scroll, you're seeing trains on the subsequent day.

tc,

On the route from Rome to Salerno, you can travel either via Trenitalia or Italo Treno high speed trains. There are trains departing both Roma Termini and Roma Tiburtina (whichever is most convenient for you).

If you're going to be in Rome for a few days prior to that trip you could certainly buy advance tickets, as long as you're willing to commit to a specific train, date and departure time. Note that your tickets will include compulsory seat reservations.

If you wish to buy advance tickets, you can buy either on the rail network websites or at www.trainline.eu at the same price.

.... "this solution is for the following day." What does this mean?.......

When you ask for the train schedule and do not specific a starting time, the web site gives you for the schedule for the next 24 hours. That often will be part of the next day since you are probably logging in in the afternoon or evening in Italy. You can only look forward. Not backwards. If you want see a specific day schedule, input 0100 (1AM) as your start time.

The solution is for the following day means that departure is not on the day you entered but on the next day. As you see solutions later in the day eventually you will reach a point when all solutions for he day have been displayed and the site will start displaying the next day departures. That asterisk is just a warning that from that point on you are looking departure the day after the date you entered.