Train Tickets- purchase now or at the station

There are three levels of train fares -- super econ, economy, basic. The first two are discount fares with limited availability and restrictions as to refund and change. THE ONLY reason to purchase tickets in advance is to take advantage of any discount fares that might be available. But that does lock you into a fixed schedule. Personally think first class is a waste of money on Italian trains. Gives a little wider seat but that is about it. And it is a short run, time-wise, to Salerno.

On the day you'll pay the Base fare travelling with www.trenitalia.com/en.html and the Flex fare with www.italotreno.it/en

Both railway companies have already uploaded September schedules, so you can check now the walk-up fares.

Economy, Super-Economy, Low-cost, Smart are all are discounted tickets that won't be available on the day of travel.

Business is a waste of money, as the proverb says: it's used only by Italian politicians and US tourists.

It’s a bit early to book train tickets now but you can definitely book them as soon as they become available, usually 120 days in advance.
One advantage of booking early is to take advantage of discounted tickets.

You can check train schedules here: Trenitalia.com

Have a wonderful trip!

Edit to add...
Thanks Dario! I stand corrected... I just checked Trenitalia and you can book your tickets NOW at great savings! 😉

If you're traveling from Rome to Salerno on the day you arrive in Italy on an overnight flight, buy your tickets upon arrival. If you buy tickets in advance, you'll need to add cushion in case your flight is late and you'll risk having to buy new tickets if your flight is really late. It would already be a long day of travel, so no sense in making it any longer than necessary.

If you're first stop is Rome for a few days, then "Yes" you could pre-purchase train tickets from Rome to Salerno to save up to 50% of the Base fare. You could also pre-purchase your tickets for the train from Florence to Venice and (it seems) Salerno to Florence. Keep in mind that pre-purchased tickets will lock you into a specific departure time.

Note that not all Freccia trains have Business Class, but all will have something similar. I only purchase these types of tickets when the pre-purchase price is equal to (or infrequently lower) than the available Second Class ticket price, but if you want to splurge then do so.

Like its been said before, you could do it either way but prices will be almost double if you buy day of. I am doing the Rome to Salerno route on my trip in May. I booked those tickets about 2 months out. And I'm getting ready to book my return trip to Rome. Safe travels.

It is not quite that dire. The difference is between 30 euro and 53 euro for a no change fast train between Salerno and Rome -- BUT note that some fast trains leave from Tiburtine. And not from Termini the main Rome station. Termini and Tiburtine are connected via subway so slightly less convenient to use Tiburtine. The train tickets are not priced like US airlines tickets that change price nearly hourly. Discount tickets are limited in quantity and the number available vary from train to train so it is possible to get a discount ticket the day of travel or the day or two before IF available. Not likely on a mid-day train but an early morning -- maybe. Train travel in Italy is one of the cheapest options compared to many of the other European countries.

I think all the high speed trains that call at Tiburtina stop also at Termini a few minutes later; anyway if you are arriving at FCO there is an airport train that goes to Termini (€ 14 pp) and another, slower, one to Tiburtina (€ 8 pp).

Tiburtina is smaller and less crowded than Termini, but the platforms are connected by elevated walkways.