Train from Rome to Pompeii.

You recognize that it will be an extremely long day. Also if you use the search box you will find responses to dozen of similar questions. Briefly - your take the train to high speed train to Naples, transfer to the Circumvesuviana train and get off at Pompeii.

But do we get to Vesuvius and then back to a train back to Rome?

All of these options are detailed in the relevant chapters of Rick's Italy book, and much of it is also in the Rome book. The national railway runs fast trains from Rome to Naples (1 hour, www.trenitalia.com). At Napoli Centrale station, go downstairs and buy a separate ticket for the local Circumvesuviana commuter line that runs to Pompei Scavi (40 min). After your visit to Pompeii, you can catch a bus from the train station to Vesuvius (allow 3 hours for roundtrip, about half that time spent on the mountain; there are also buses to/from Ercolano Scavi station). The reverse by Circumvesuviana and Naples-Rome train.

Forget to add the bus option. Obviously, that is correct. But seriously rethink going to Vesuvius. There is nothing there except a hole in the ground and a lot of loose to climb through to rim of the crater. Unless it high on the list of been there, done that, it is hardily worth the time and effort. Besides, Pompeii is enormous and can easily consumer the whole day. If I had time left over, I would spend it in the National Museum in Naples which is where all the good stuff from Pompeii is displayed. And there is the Secret Room.