There is luggage storage in the Naples train station and at the entrance to Pompeii. IMO (and I'm much older than you), the archeology museum is an easy walk from the Naples train station. Herculaneum is an easy walk from the Circumvesuviana train station.

You would have to investigate how to get to Vesuvius, although someone here will probably chime in. I'm sure there's a bus from somewhere.

How long in each place? Nearly impossible for anyone to answer. On our first visit to Pompeii we spent the better part of an entire day from opening to closing. (If it had been hot and humid, I wouldn't have lasted nearly as long.) Other people say they've seen enough after a couple of hours. I found Herculaneum less interesting than Pompeii; others disagree. (I would happily give up Herculaneum in order to spend some time on Capri or see a little of the Amalfi Coast.) We spent a half day at the archeology museum.

Herculaneum and Mt. Vesuvius will be a problem on travel days as there is nowhere to store luggage.

The museum is closed on Tuesday so you would have to visit it on the way down, not back. If you left Rome early you could store your luggage at the Naples train station, visit the museum and then perhaps visit Herculaneum (you would have to train back to Naples to retrieve your luggage afterwards). Or it might be possible to catch a bus from Naples to Vesuvius and then retrieve your luggage.

If you do try to visit the museum and Herculaneum on the same day, you might want to get the 3 day Campania Arte Card Tutta le Regione for 27€. It would include admission to both and trains/buses/metro/trams for three days. It would expire before your return though. You can buy it in the Naples train station.
http://www.artecard.it/#

I understand that you will have a hotel in Sorrento for 3 nights, isn't? If yes, I think that you can travel by train from there (without luggage) to visit Pompeii, Herculaneum, even Naples.
Not sure about Vesuvius.

Too bad there's nowhere to store luggage at Herculaneum or Vesuvius. Oh well. We're planning on Capri and Amalfi Coast towns our two full days in Sorrento. Just trying to see which of these other "bonus sites" could be feasible for the travel days.

Luggage aside, has anyone climbed Vesuvius? Is it convenient to get to from the train stations? Worth the climb?
Thanks!

To get to Vesuvius from the train station you have to take a special van that departs not far from the station. You also have to take the exact same van down after an hour or so or you will be strongly reprimanded. We enjoyed the climb and great views. For more info see:
http://www.fodors.com/community/europe/bruce-and-marija-go-to-naples-alba-neive-turin-florence-and-paris.cfm

Herculaneum is well worth the time and effort to visit. I found it more appealing in many ways than Pompeii (mainly because there were so many fewer visitors.) Leave your luggage at your hotel one day to go to Herculaneaum (easily reached as described by Jean)--it's worth it.

If you're strapped for time and staying in Rome, take a very quick (and very worthwhile) trip to Ostia Antica. Best advice we were given when staying in Rome and longing to visit Pompeii, but couldn't due to lack of time. It was absolutely brilliant.

NatKP that sounds interesting, how did you get there?

Try this - it will tell you all you need, and lots more!

http://www.ostia-antica.org/touristguide.pdf

Peter