Short trains from Rome to Naples, Pompeii, Sorrento

Rome to Naples is a high speed train. Tickets are for a specific train. You can buy at the station on the day, but then you pay full price. It would be cheaper to buy in advance. Buying in advance also ensures you get 4 seats together.
You can see times and prices, and buy tickets, on the Trenitalia website: https://www.trenitalia.com/en.html

The other two legs (Naples to Pompeii and Pompeii to Sorrento) are regional trains. No reserved seats and no discount for advance purchase.

You will definitely wish to take a fast train from Rome to Naples either Trenitalia , or from the rival company Italo. These make the journey in somewhere around 75 minutes, whereas a normal "Slow" train can be in excess of 3 hours. These need booking in advance, and as a general rule the earlier you book, the cheaper it will be.

I'm not even certain you can make reservations on the circumvesuviana, and there would be no advantage in doing so,

Not really a question of the train selling out- highly unlikely. The train from Rome to Naples would be a high speed Freccia train. You could save money by buying in advance if you can commit to a specific date and time. Otherwise buy on the day. The Circumvesuviana or the Campania Express are local trains and don't need reservations. If the seats are full, you stand, just like on a subway or LRT. Buy at the station.

There are no reservations on the Circumvesuviana and it is cash only

You can reserve the Campania Express-

Here is info:
https://www.sorrentoinsider.com/en/campania-express-train-schedule

buy tickets here:
http://ots.eavsrl.it/web/public/ots/ticket/seleziona-andata

Thanks to all of you for your responses. I appreciate it.

I went onto Italotreno and they have a train that goes from Roma Termini to Napoli C.le. Is Napoli C.le the destination that I want if I then want to catch the circumvesuviana train to Pompeii?

Yes it is.

Thank you

When I take a train back from Naples to Rome, does anyone know if I can take a train directly from Naples straight to the airport in Rome? Or should I take the train to Roma Termini and then another train to the airport from there?

First, note that Italo and Trenitalia are two different companies. Both run high speed service on the Naples to Rome route. Either one is fine, if one has a better price or better timings for you.

"When I take a train back from Naples to Rome, does anyone know if I can take a train directly from Naples straight to the airport in Rome?" No, you have to take a train to Termini, then the Fiumicino Express from Termini to the airport (separate tickets).

Thank you so much.

"No, you have to take a train to Termini, then the Fiumicino Express from Termini to the airport (separate tickets)."

Or if the train goes to Tiburtina (which is one of the stops for Italo, and some Trenitalia Freccia trains), then change to the local airport train from there - slightly slower but about €5 cheaper. It's been some time since I did this, so my information might be out of date.