Rome to Sorrento for 3 nights, then back to Rome - how to best do Pompeii?

You can absolutely do this without renting a car; many tourists do! You take the Circumvesuviana commuter train or Campania Express train from Sorrento to Pompeii and check your bags at the excavation. Then take one of those trains onward to Naples, and a fast train from there to Rome.

Pompei Scavi - Villa dei Misteri station is right outside the main entrance to the excavation. Buy another ticket at the station to get to Napoli Garibaldi, which is connected to Napoli Centrale for picking up fast trains to Rome.

https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule?timetable_route=sorrento_napoli

The Campania Express has fewer trains but more comfortable carriages:

https://www.sorrentoinsider.com/en/campania-express-train-schedule?timetable_route=sorrento_napoli

Wow thanks Kathy! How do we check our bags at the excavation?

There is information at this website about leaving your luggage, either at the Pompeii Scavi train station or at the Pompeii site:
https://www.earthtrekkers.com/visit-pompeii-traveling-rome-sorrento/

Note that the Pompeii train station guide indicates there is NOT luggage storage at the train station:
https://www.italiarail.com/train-station/pompei-train-station-guide

but readers firmly declare they have used it...as recently as this January. It's certainly worth checking because the luggage storage at the Pompeii site can fill up early.

Perfect, thanks! I think we should be okay if we leave early from Sorrento so we'll get to Pompeii early. Also, looks like it would be fun to take a ferry or hydrofoil from Naples to Sorrento instead of the train or car!

No firsthand experience checking luggage as we day-tripped from Sorrento but I'd read the same about available storage at the station.

Storage at the scavi is free but can fill to capacity when it's very busy.

it would be fun to take a ferry or hydrofoil from Naples to Sorrento
instead of the train or car!

Some folks do it that way, It's just not as convenient as you have to transfer from the train station in Naples to the port, and then transfer UP the cliff from the marina when you reach Sorrento

The convenience of the commuter or express train is that its station is connected to Napoli Centrale so is just a short walk from one to the other. The train station in Sorrento is in the upper part of town and not a long or difficult walk to some of the hotels (true for ours but not all). Anyway, no right or wrong to it, just personal preference regarding # of transfers one is willing to make.

(Edited out duplicate link to the earthtrekker's site provided by CWsocial above)

Note that the Circumvesuviana is a very crowded, sometimes hot, sometimes infested with musicians collecting donations, Metro-like train. But it's very cheap and very fast. I think taking the ferry is trying to pack too may things into 3 days, but it is perfectly practical. You're the ones humping the luggage! (And it can be hard to fit the luggage onto a Circumvesuviana train that is already over-capacity with standing people.

You may wish to Search (top center) for past discussions over SITA busses versus expensive car and driver companies for your day to the Amalfi Coast. You do not want to be driving your own car there - the traffic is awful, and there is virtually zero street parking.

I think this thread plays down the large and well-organized luggage storage at the Porta Marina (and other) entrances of Pompeii Scavi, the actual excavation. I have never read a negative remark about using these live-staffed counters. The Porta Marina entrance is about 250 feet from the Pompeii Scavi stop of the Circumvesuvina.

Even in mid-May, it can get very sunny and hot in Pompeii. Be prepared with hats and water. There is little or no shade, although there is an air-conditioned restaurant.

I think this thread plays down the large and well-organized luggage
storage at the Porta Marina (and other) entrances of Pompeii Scavi,
the actual excavation. I have never read a negative remark about using
these live-staffed counters.

Tim, no one "played down" or said anything negative about the luggage storage facility at the scavi. I just mentioned that it can fill to capacity if the excavation is very busy. That's stated on coopculture's website as well as some others. It's just a head's up that if it's full, storage at the station is another option.

https://www.coopculture.it/en/heritage.cfm?id=1

Visitors can leave their bags and personal belongings at the free
locker room at the access gates. However, during some periods the
sites are very crowded and the locker room may not contain the baggage
of all visitors,

There is luggage storage at the Pompeii Scavi station. It's located downstairs in the tunnel under the track. I'd check it here rather than shleping it to the main gate.

The station Italiarail mentions is the Main Pompei station that is located between Naples and Salerno. No luggage storage there.