Rome to Pompeii

You can take the freccia train from Roma Termini to Napoli Centrale, go downstairs to the Circumvesuviana station, take the train in the direction of Sorrento, get off at Pompei Scavi. This will take at least two hours, probably more. You can hire a guide onsite. Allow lots of time, the archeological site is huge.

You may want to stop in Napoli on the way back to visit the Archeological Museum and have dinner along the waterfront before you go back to Roma.

I'd also look into the possibility of a private driver for the round-trip, either from Rome or, better, from Naples. You could take the local Circumvesuviani train from Naples but you could spend a lot of time just waiting for the train in each direction and then waiting to board your train back to Rome from Naples. Take into account that you cannot rely on the promptness of the local trains. I used them in low season (February!) when there were few locals taking them and even fewer tourists and they were invariably behind schedule.

It is logistically easy. Once in Napoli Centrale, there is good signage to the Circumvesuviana lines. Be flexible with the Circumvesuviana-as Chani said, it can move slower than expected and/or be delayed. The first time I took it, it took about 2 hours to go from Naples to Sorrento (which is longer than it is supposed to take) and that was NOT counting the 30 minutes of time we lost because the train broke down. I think Pompeii is about halfway to Sorrento which is the end of the line. But it is definitely the cheapest way to get from Naples to Pompeii. And on a day where the trains are running well and traffic is bad, it may be shorter than driving. And it is a short walk from the station to the entrance.

If you know what day you are going, you can save a lot of money by booking the Rome-Naples legs of the journey in advance and the trip is about 1 hr 15 min if you take the fast train.

I understand not wanting to do a big tour but I agree with Zoe about hiring a guide once there. I tried to do it on my own and found it pretty overwhelming, in large part due to the size of the city. It would have been better to have some direction.

I second what several other posters have said above.

Pompeii is a very manageable day trip from Rome. Just allow time on both ends for the train ride (2hrs or so)
A private driver would be nice but not necessary really. Much cheaper to just take the train.

I do highly recommend a guide. It was my 1 true regret on my trip to Pompeii. It is such a huge site and after we returned home I realized we'd missed many of the most beautiful parts of the ruins because we didn't know where to find them (even with the map and a decent sense of direction). At the very least, bring Rick's Audio Europe app with you and listen as you explore. :)

I'd allow 3-4 hours to see the ruins in total. You can also visit nearby Herculaneum or climb Mount Vesuvius in the same day with a lunch break between. Its a long day with the addition of one of the other locations but we managed to see Pompeii and Vesuvius in 1 day and still arrive back in Rome before dark. :)

I will be doing the same thing with my boyfriend in couple weeks. You might want to check out the Mondo/Rick Steves shared tour (https://www.sharedtoursmondoguideforricksteves.com/), the tour meets at 1030. We booked the 0735 train from Rome to Naples gets in to Napoli Centrale at 0845. When are you going? I can report back.