Sorrento

Yes you surely can.
The Circumvesuviana train (or a boat) will take you to Naples. There is a bus that takes you all the way to Amalfi, stopping in Positano and a few random stops along the way. Once you get to Amalfi, you can change buses and go visit Salerno, which has more of a "city" feel. In Salerno, there are other buses that can take you elsewhere, i.e. Paestum to see the ruins. You can get a 24 hour bus ticket from 6 euro (this was last year's price) which is your best bargain, although I think you may need to buy a separate ticket for Paestums since it's a different bus company. The Cirumvesuviana train and the buses can all be caught at the station in Sorrento center, a nice walk from many of the hotels.
Happy travels!

Agree/don't rent a car/use bus/train they are all from the from the same area can go via bus to amalfi o positano and trian back to pompeii enjoy also great tiny/tiny pizza place right by train/bus station best pizza we had walk down from station take first left and walk about three stores down on the left enjoy trip

We have never been to Southern Italy, thank you for your advice. Our plan is to fly into Rome. Can I then take a train to Sorrento and use as a home base for day trips? The next week we want to return to Rome for a couple of days, before boarding a cruise out of the port near Rome. Any advice? Is this all logical?

Yes take a train from rome to naples and then to sorrento fast train hour from rome slow train much cheaper but twice as long but all of us liked it better depending on time of day 10 euro versus 40 euro train from naples to sorrento about an hour maybe 10 euro but not sure use sorrento as base and take day trips to pompeii/the coast etc do not attempt to drive use bus get on and off as you please and leave the driving to them

That's right. You cannot take the train directly from Rome to Naples, so you will get off in Naples at which point you can either take the Circumvesuviana train (as mentioned above) or you can take a boat across the bay. We took the boat going to Sorrento (breathtaking views) but it does add issues getting to the dock. If you choose to take the Circumvesuviana train to Sorrento, you just basically take the escalator down a level from where you get off of the train in Naples. It depends on what you want to do.

In Naples, the Alibus does a loop between the airport, the main train station/Piazza Garibaldi, and the port (Molo Beverello). They leave every half hour and you can buy the 3 euro ticket onboard. It's a relatively easy way to get from the train station to the port so you can take a boat to Sorrento.