Sorrento to Rome

The Circumvesuviana commuter train from Sorrento to Naples would be a lot cheaper. There have been a lot of horror stories about it that seem to scare people, but I thought it was just fine. Very basic for a train but perfectly fine. Pay attention to your belongings, don't get careless. Get to the station in Sorrento early so you can get good seats when people can first board the train.

Why not return the way you are going to Sorrento?
Otherwise: The cheapest option is to take the Circumvesuviana to Naples Garibaldi and then walk next door to Naples Centrale Train Station and catch a train to Rome
http://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule

A cheaper option and more comfortable than the Circumvesuviana is to take the Campania Expres
http://ots.eavsrl.it/web/public/ots/ticket/index

to Naples Garibaldi and then walk next door to Naples Centrale Train Station and catch a train to Rome

And yeah on the Circumvesuviana pay attention to what is going on. On one of my journeys on it last month an American tourist was not paying attention and her IPhone was gone when the train reached the end of the line at Sorrento.

We hired a private driver from Sorrento to Naples, then took the train from Naples to Rome - NEVER again. Next time, we will take the train from Sorrento to Naples, then from Naples to Rome. Traffic on the road from Sorrento to Naples was terrible - truly a parking lot. Our driver said that is normal. We missed our train in Naples to Rome (we had purchased tickets here at home for less money) and had to purchase them at the station for a lot of money. And yes, we gave ourselves 2.5hours to get from Sorrento to Naples by car, and still missed the train. Do yourself a less stressful favor, and just take the train. We used the train while in Sorrento to get to Herculaneum - watch your bags, but it was fine. Never heard of the Campania mentioned above, but seems like a great option.

The Campania Express takes 30 minutes. Sorrento to Naples.

Yes, the Campania Express train can be an option to the regular (and very inexpensive) Circumvesuviana train from Sorrento to Naples - from where you'd take a fast Trenitalia train to Rome - but it has very limited daily runs.

https://www.eavsrl.it/web/content/campania-express-2018-it

The Campania express actually takes 48 minutes from Naples to Sorrento and has to be convenient to your schedule. It appears there are 6 trains a day that run to Sorrento.

I have used the Marozzi bus in both directions and found it preferable to the Circumvesuviana/Napoli train. The bus takes about 4 hrs to Rome, arriving at Tiburtina station. It makes a stop along the way for a quick bathroom break at one of the rest areas on the autostrada. Luggage is stored down below. Here's a link for you to check the schedule for your date: https://www.marozzivt.it/home.

Buon Viaggio!