Traveling from Rome to Naples

You can do a day trip from Rome [a longish one] to Pompeii or Capri, but not both. Both you would have to stay down there overnight.

Pompeii you take a train to Napoli Centrale, the main station, then downstairs to Garibaldi and the commuter train to Sorrento, getting off at Pompeii Scavi:

https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule

Capri you take a ferry from Sorrento or Naples.

According to rometorio.com, the only way from Capri to Pompeii is via ferry to Sorrento. After you get off the ferry, you have to make your way up to the train station - a lot of stairs or an elevator (there may be a wait) up to the town center, then a 10-15 minute somewhat uphill walk to the Circumvesuviana train. There may be a crowd at there to board the train - there are about 2 trains an hour and it's about 1/2 hour ride. There's also the Campania Express at 10.19 that is more comfortable and a little faster - though it could also be crowded in mid-September. Consider taking a taxi from the ferry to Pompeii if you don't want to spend a night in Sorrento - or at least taxi from the ferry to the train station. From Pompeii to Naples train station is another 40 minutes on the Circumvesuviana, then you need 10-15 minutes (I'd allow more) to go up to the main station, find your platform and board the train to Rome.

This was in December 2012 so a few years ago and a different time of year, but after dropping off our rental car in the center of Sorrento, we took the Circumvesuviana train to Naples, purchasing tickets just before boarding, then bought tickets for the next train to Rome, arriving mid afternoon. The Circumvesuviana train was almost empty when it departed Sorrento, but got fuller and fuller as it neared Naples, to the point that it was standing room only the last couple of stops before Naples. And there was one pushy, nasty man towards the end of the line in Naples, so keep an eye on your baggage and don’t get up if you want to retain your seat..

We’d visited Capri from Sorrento a couple of days earlier, so didn’t have the ferry from Capri to try to work in to the journey that day. Not sure what the schedules are these days, but it seems it should be possible to reach Rome from Capri in a day, as long as you’re not planning any sightseeing along the way.

I agree with the route Chani provided.....

• take the Ferry from Capri to Sorrento
• make your way to the rail station and buy a ticket to Pompeii Scavi on the Circumvesuviana (it can be a bit dodgy at times, so watch your valuables). While the Campania Express is also an option, I don't believe it runs as frequently.
• When you arrive at Pompeii, I believe there's a place you can leave your luggage while you're touring the site (check the guidebook for details).
• When you're finished touring Pompeii, retrieve your bags and buy another ticket on the Circumvesuviana to Napoli Garibaldi.
• When you arrive in Naples, walk upstairs to Napoli Centrale and buy a ticket on either the Freccia or Italo to Roma Termini. While you could buy that ticket in advance, it's probably better to pay a bit more and buy it at the time, since you don't know how long it will take to tour Pompeii and get to Naples. Keep in mind that tickets for the fast trains are specific to train, date and departure time, so if you miss the train listed on your advance purchase ticket, the ticket is worthless and you'll have to buy another one anyway.