Rome to Pompeii/Herculaneum with overnight in Paestum?

Why not go straight to Paestum from Rome and work your way back instead of trying to pack Pompeii and Herculaneum in one day, especially if you have your baggage with you. Pompeii has storage but not certain about Herculaneum.

Pompeii and Herclaneum in one day is extraordinarily ambitious. Starting the day in Rome makes it borderline lunacy. Better to do one, and do it well, than to jog through both of them. Trying to add Peastum in with either is an exercise in futility.

Slow down and look at what your transportation logistics are and I think you will see what you are up against is really not practical.
Also check what the daylight hours are for when you will be there.

Herculaneum, Pompei and Paestum in the same day is just not feasible.

What you can do is to visit Pompei and Herculaneum, then heading to Paestum just for one of the nice rural-setting hotels around Paestum (pick something close to the ruins, not by the sea, i.e. walking distance). Even though, arriving after sunset at Paestum and going to the hotel would require a bit of advanced-planned logistics.

To complicate things further, the railway station in Pompei that leaves you closest to the ruins entrance (Pompeii Scavi/Villa dei Misteri) is not on the same railway that connects Napoli to Salerno (and Paestum further south), so it would require that you go from Napoli to Pompeii on the main rail line, stopping at Pompei FS (the main town station), walking to the other ruin area entrance, leaving your bags there and then going back to take a slow train to Salerno and Paestum.

Summing it all: unless you have one extra night, I'd cut out Paestum, pick a hotel in Sorrento or Vico Equense, and head to either village after your tour to Herculaneum and Pompei. Then you catch a ferry from Sorrento to Capri. You really need two nights between Rome and Capri to visit the three sites.

Here is an alternative arrangement if you have one extra night:

1. Take a train from Rome to Salerno (most of it will be high-speed until Salerno, avoid slow and much longer routes) early in the morning. Set yourself up at a hotel near the train station (there are some comfortable and affordable options and the area has been renovated/improved a lot from its decay of 10-15yrs ago). Leave the bags and catch a train to Paestum. Visit Paestum in the afternoon (the ruins are just 300m from the train station). Take a train back to Salerno.

2. Take a train to Pompeii FS, walk (~12-15min) to the main entrance of Pompeii. Visit the ruins (they are huge), then exit via Porta Marina (the other entrance near Villa dei Misteri, served by EVA trains to/from Napoli and Sorrento). From this other train station (Pompeii Scavi/Villa dei Misteri), take a short train ride to Ercolano Scavi. Visit the site. Once again, walk to the other station in town (Ercolano FS), where you catch a train back to Salerno.

3. After a good night of sleep, take the one of the many ferry options to Capri from the Salerno dock. This trip will also give you a very nice sea view of the Amalfi coast!

For this to work you will need several train trips
Trenitalia - Rome-Salerno (buy in advance for cheap fares, beware of cheaper but much longer routes, buy the high-speed 'Freccia' routes)
Trenitalia - Salerno-Paestum (regional trip, fixed prices, no reservation)
Trenitalia- Paestum-Salerno (regional trip, fixed prices, no reservation)
Trenitalia- Salerno-Pompeii FS (regional trip, fixed prices, no reservation)
EVA (former Circumvesuviana) Pompeii Scavi-Ercolano Scavi (regional trip, fixed prices, no reservation)
Trenitalia- Ercolano FS-Salerno FS (regional trip, fixed prices, no reservation)

Hope I helped give you some alternatives. The train trips are all very easily done, all the regional routes have 1 to 4 trains per hour during daytime. You can buy regional tickets in loco. The one from Rome-Salerno, in advance, online.

My wife and I are going to that area this spring. Here's a page where you can download the PDF guide to Pompeii.
The "Pompeii" "guide" link is to a long pamphlet style document that describes all the buildings. They have some suggested routes for those who want to spend 2 hours, 3 hours ... 7 hours.
One way to help you go around the site fast may be to hire one of the guides at the gate. I've read that they give 2 hour tours.
That kind of discipline may be needed if you're really trying to cover a lot.

http://www.pompeiisites.org/Sezione.jsp?titolo=MAPPE%20E%20GUIDE&idSezione=6844

Thanks for the advice folks! We definitely aren't planning to tackle Rome as a part of all of that. We are giving the city its own dedicated days. Then the thought was a day taking the train from Rome to Pompeii/Herculaeum, then going on to Paestum to stay in one of those nice country hotels someone mentioned for that night. Then we would actually see Paestum in the morning, and taking a train back to Salerno that same day and going on to the Amalfi Coast from there for a number of days.

The other idea, would be to do our time in Rome and then - over several days - hit Sorrento, Capri, Positano, Amalfi and end at Paestum before taking the train back to Rome. Looking at all of your responses, this may be the less arduous track.

Thanks for all your help!!

You are likely to get better answers and suggestions if you told us how many nights you have for this trip and where you plan to go and where you have to be to get your plane home

A "few" days here and there is a pretty vague target