Train from Salerno to Rome

it would be cheaper

We did last year. We saved some money travelling on one of the quicker trains, but you don't have to. There are plenty of trains going throughout the day.

If you want to save money and can accept tickets that are non-refundable and non-exchangeable (or will have severe limitations on these actions), buy in advance online. If you need full flexibility and don't mind paying full Base price, you can buy them when you get to Italy, from any station in the country (you don't have to wait until you're in Salerno).

To find all your options, go to Trenitalia's website: http://www.trenitalia.com/tcom-en. For the stations, use "Salerno" and "Roma (tutti le statzioni)," which means "Rome all stations" (even on the English version of the website you need Italian city names). Looking quickly, I see that for second class on the Intercity train (takes about 3 hours), the SuperEconomy fare (totally non-changeable and non-refundable) is €12.90, the Economy fare (no refund, but you can apply its value towards another ticket if you change it before travel) is €19.90, and Base fare (totally exchangeable and refundable) is €30.50. For the standard class on the Frecciarossa train (takes about 2 hours), the SuperEconomy fare is €16.90, the Economy fare is €34.90, and Base fare is €50.

So, in a way you can't lose. On the one hand, there is a substantial saving on Super Economy tickets. On the other hand, Base fare on this route won't break the bank, if you need or want full flexibility, or or if the cheaper tickets are sold out for your dates (they are limited in number, and once they're gone, they don't come back).