Pompeii and Vesuvius from Sorrento

We used the Circumvesuviana to and from Naples and Sorrento to/from Pompeii. Easy. There is luggage storage at the Pompeii train stop.

is Campania Express Train better than Circumvesuviana?

Depends on what "better" means. The issue I have with the Campania Express is the infrequency of its runs: only 4 times daily, each way, between Naples and Sorrento. The Circumvesuviana, while an unlovely commuter train with zero creature comforts, makes many, many more of those runs daily.

https://www.sorrentoinsider.com/en/campania-express-train-schedule
https://www.sorrentoinsider.com/en/campania-express-train-schedule?route=sorrento_napoli#section-schedule-result

https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule
https://www.sorrentoinsider.com/en/naples-to-sorrento-train-schedule?route=sorrento_napoli#section-schedule-result

What about the best method to transfer from Pompeii to Vesuvius and back? I know there are so many tour options with transport included. And which train station into Pompeii is easiest to get to the ruins from?

Month and Year?

https://community.ricksteves.com/travel-forum/italy/pompeii-and-vesuvius-69e8366c-1e75-4a14-a0bc-0677269492d1

Many people find the commercial shuttle busses convenient. We had no interest in sweltering up Vesuvius, having been to Mt St Helens and Hawaii. We preferred seeing Pompeii from our Sorrento hotel, but the free Pompeii ticket office bag check is very reliable, if your shuttle bus returns there.

And which train station into Pompeii is easiest to get to the ruins
from?

From Sorrento, Circumvesuviana to Pompei Scavi-Villa dei Misteri station. It's right in front of the main entrance to the scavi. The schedule is in one of the links I'd previously posted. This train also runs to Naples. There are bag-check services at the station or at the scavi itself.

is Campania Express Train better than Circumvesuviana?

I thought the Circumvesuviana was underrated and the Campania was overrated. The Circumvesuvia is nothing pretty to look at, but for the 35 minute ride from Sorrento to Pompeii, we never had any issues, and it's a 3rd of the cost. We unknowingly rode the Campania a couple of times and didn't realize it. It was standing room only for us from Herculaneum to Sorrento one day and from Sorrento to Naples on another. There are overhead luggage racks, but they were to small for my carry-on suitcase which came falling down onto my lap during the trip.

If you are boarding the Circumvesuviana in Sorrento you will boarding an empty train as the line begins there
We had no issues with it on our ride to Pompeii

The return was a bit crowded but we were able to find seats

What about the best method to transfer from Pompeii to Vesuvius and
back? I know there are so many tour options with transport included.

(Full disclosure: it has been a few years since I went to Vesuvius, and things may have changed since then.)

We went to Vesuvius on a whim at the end of our group tour of Pompeii. We simply asked at the Scavi station how to get to Vesuvius, and they pointed us to a ticket office around the corner. My recollection is that there was a regularly scheduled service that ran every 1/2 hour or maybe every hour (?). We waited at a bus stop across from the Scavi station. In about 20 minutes a bus showed up and took us to the entrance to the national park. We got off the bus at the entrance and got on a big all-terrain bus (with huge tires) that took us to a drop off area not far from the summit. My recollection is that we had to have a second ticket for the big bus. I do not recall whether our original ticket included a voucher or whether we bought a separate ticket at the park entrance. It was a very bumpy ride from the park entrance to the summit, and we had to hold on to our seats. Once we got to the drop off area, we hiked 30 minutes, maybe less, to the summit. I don't recall being under any schedule pressure to catch a bus back down the mountain.

Once we arrived back at Scavi, we took the Circumvesuviana back to Sorrento.