Best way to get from Rome to Pompeii for day trip in June

Fast train from Rome to Naples. Change there to Circumvesuviana train and get off in Pompeii. It is not a good idea to see Sorrento in the same day. Good idea would be to stay overnight in Sorrento and visit Pompeii from Sorrento.

That makes for a pretty long day. Not impossible, not entirely unreasonable, but longer than I would probably enjoy (especially in June, when it's going to be pretty hot). Be sure to pace yourself and stay hydrated.

Read what David said..... and have no big plans for the following day.

Is it a good idea to try to see Sorrento in the same day?

NO!

As Ilja posted: fast train to Naples, commuter Circumvesuviana train to Pompeii.

I'd choose doing it solo by train versus a bus tour as it takes much longer to get there via road, and most of those tours don't spend long enough at the excavation before moving you along. It's a VAST site that can - and did for us, anyway - eat the better part of a day if you find it very interesting. You can get tours at the entrance if you wish, and then spend some time exploring on your own before heading back to Rome.

No, I wouldn't recommend Sorrento on the same day either. I'd overnight there if wanting to do that one.

Sarah Murdoch's day trip https://adventureswithsarah.net/travel-tutorial-day-trip-rome-to-pompeii/

I'll always recommend a guide to get a better sense of what you're seeing. 'Through Eternity Tours' can be hired out of Rome.

I am so grateful for all of your advice. Thank you everyone for your recommendations and time.

Please don't try this...go to ostia antica if you want old ruins while in Rome

I second Ostia Antica, especially if time is of concern. Ostia Antica is less than an hour out of Rome via train. It is an entire ancient city that has been discovered, smaller than Pompeii but still very interesting. You can go early and see all of it in half a day and be back in Rome by mid afternoon.
If you do go RSteves audio tour is a good choice.

First off, don't think about going to Sorrento the same day.

But yes, Pompei can be done on a daytrip from Rome, and quite inexpensively to boot. If you have time to play with, a cloudy, cooler day would work much better than a hot one. And seeing that a do-it-yourself trip does not require reservations of any kind, conceivably you can choose your day depending on the weather. Oh, rain is not good either because those stones at Pompei get mighty slippery!

Leave Termini station on a Freccia train, certainly before 10AM. That would put you down to Napoli Centrale station no later than 11:30. I believe you walk downstairs to catch the Circumvesuviana commuter train, which will take you to the Pompei Scavi station, about 4-5 stops away if memory serves.

Once there, walk the short distance to the entrance to the ruins. There will be random guides wanting your business, and sometimes there are good ones that can be bargained with for, say, 20 Euro for 2 hours. Seeing that the ruins are massive and not really marked in any way, shape or form, a guide could be beneficial. When we were there in March 2015, I had done a lot of advance historical research and we used the RS Pompei podcast in real time, which was pretty sweet. But in hindsight we should've hired a guide to be more efficient. In 3.5 hours, we saw by my estimation 70% of the ruins.

Then simply re-trace your steps. You could take a taxi from Napoli Centrale to sample one of Naples' famed pizzas in between on the way back. Either way, you get back to Roma Termini early evening at worst. Yes, a lot of walking but that's expected when touring.

Enjoy your planning!