Itinerary Question

You could train from Rome to Naples and spend two nights there, giving you a full day to see some of the sights like the archeological museum. Then take the train to Sorrento and stop off at Pompeii on the way, which would save you the extra transportation involved in a day trip. My initial reaction is that staying in Sorrento and Positano and Capri is one too many hotels when they're all relatively close together, but others might have better feedback there.

On your extra day, you could visit Herculaneum. It's right on the train line between Naples and Pompeii and it gives a different perspective of the Vesuvius eruption. Or you could go to Paestum or Oplontis. (This is assuming you're interested in archaeological ruins, which I realize may not be the case.)

https://www.italyreview.com/villa-oplontis.html

https://www.sitiarcheologiciditalia.it/en/paestum/

We have been to all of those places! You will have a wonderful trip. You may wish to change the order of your locations slightly. Sorrento, Pompeii, Capri , then Positano and Amalfi Coast, then Naples, then Rome. Seeing Pompeii before you see the National Archaeological Museum in Naples is helpful. Or you could begin with Positano and the Amalfi Coast and after that go to Sorrento, Pompeii, Naples, Capri. We especially loved Sorrento and stayed near Villa Communale in a hotel with a view across the Bay of Naples. Sunsets are gorgeous there!

This is a more efficient itinerary.

Rome-Salerno (1H31m). Salerno-Positano ferry (1H20 m). Positano-Sorrento ferry ((45m). Sorrento-Capri ferry (20-30m). Capri-Naples. Ferry. (1+hour).

Edit: Ferry timetable.

https://www.travelmar.it/_download/orari-travelmar-aprile-2022.pdf

Look carefully at Gerri's plan.

It doesn't sound like you've taken into account how ground travel works. From Rome airport, you need a train to Rome (Termini), then a train to Naples, then a local metro train - Circumvesuviana to Sorrento. From Capri, you need a ferry to get to Naples, then transportation to the train.

I like Gerri’s plan a lot. It doesn’t have you hauling luggage on the Circumvesuviana train from Sorrento to Naples which is a safer way to travel on that train. It’s fine if you’re not carrying valuables so a day trip to Pompeii from Sorrento or Naples works fine although Naples is closer to Pompeii.
Sorrento is where I stayed when I visited the Amalfi Coast and the only thing I liked about it besides being a great base to explore was the passeggiata when families take their early evening stroll and eat a gelato.

To add to my post above, keep your Naples day and Pompeii day at the end. Do Pompeii early in the day, and the archeological museum in the afternoon.