You can reach Herculaneum by Circumvesuviana, a local commuter train connecting Naples to Sorrento, and stopping at all the important achaeological sites in-between. Tickets are priced depending on the length of your journey, and are valid for a certain number of minutes, including return trips (if they are within the time limit). Buy your ticket directly at the station, and if there is time remaining at the end of your trip you can use it to connect with local city busses. You can see ticket prices at the site below.
Main Stops: Napoli Centrali Train Station (lower level), Pompeii Scavi (archaeological site), Ercolano (archaeological site), Sorrento (via degli Aranci, five minutes by foot to city center).
Many people report having purses or wallets stolen on this train. I personally did not have any problems, nor did my family when they visited. It's always best to keep your hands on your purse at all times, and to keep your valuables where wandering eyes (and hands) can't see them!
From the main train station in Naples, buy a ticket on the Circumvesuviano train for a small fare and travel to Ercolano (Herculaneum). (Ignore the little old man who wants to buy your ticket for you; he'll want money.) From the Ercolano station, walk down the hill, following the "Scavi" signs and you'll be at the archeological site. Stop at one of the little stores and pick up a bottle of water Ask for directions, even if you don't need them. The clerk will like you asking.