This is dead easy to do on your own. We skipped Vesuvius as you can see the thing clearly from Pompei and we just didn't see the need to make that that trip. The scavi of Pompei is vast and took the lion's share of a day to do...and we still didn't get it all done.
By (cheap!) Circumvesuviana commuter, it takes about 35 minutes or so from Napoli Garibaldi station (connected to Napoli Centrale)to Pompei S. Villa Misteri station, across from the main entrance:
Going to Pompei:
http://www.eavsrl.it/web/sites/default/files/eavferro/Napoli%20-%20Sorrento_1.pdf
Going from Pompei back to Naples:
http://www.eavsrl.it/web/sites/default/files/eavferro/Sorrento%20-%20Napoli_0.pdf
As you can see from the schedules, Ercolano Scavi is also this route.
The Campania Express train, as mentioned in a previous post, is the other option but doesn't run as frequently:
http://www.eavsrl.it/web/content/campania-express-2017-it
Important: if taking any of the trains above, validate your tickets before boarding! It's not unusual for on-train agents to make random checks on these routes, and fines are high if caught with an unvalidated ticket. There is NO leniency for the tourist who claims they didn't know any better, and you'll be expected to pay the fine on the spot, no argument.
Validation of tickets is necessary for virtually every train in Italy for which you don't have a reserved seat.
Back to Pompei Scavi: you will need some sort of audio or printed guide and map to find your way around and know what the heck you're looking at. We bought a book at the entrance, that I've referenced many times since, but an audioguide might be fine for you (rent there or download before you go). Here's a pretty good map that'll give you an idea of size/layout of the scavi:
http://www.pompeiisites.org/allegati/Pompei_120515053335.pdf
And a website which may be helpful; lots of info on the building (you may have to pull down the tab for English more than once):
http://www.pompeionline.net