Traveling by train is a great way to experience Europe. This is how we often travel.
If you are visiting the Netherlands and you want to travel by train, anywhere within the country, then you will want to visit this website. This is where you can find up-to-date train schedule information, applicable fares, etc.
http://www.ns.nl/en/travellers/home
The train also stops directly at Schiphol (Amsterdam) airport, bottom level. This makes it very easy to transfer from train to plane and vice-versa. (The train ticket booth can be found in the main lobby of the airport.)
In Den Haag we have at least four train stations with the two biggest of these being Den Haag Central and Den Haag Holland Spoor. We often use Den Haag Holland Spoor when we are traveling to/from Delft, Rotterdam, Maastricht, etc. Den Haag Central is where we usually go in and out of when traveling to/from Schiphol airport, Amsterdam and cities to the North.
Both train stations have easy access to Den Haag's local trams, which are an easy way to travel about within the city. (In fact, tram 17 will drop you off directly across the street from our home. Just saying... for our family and friends who are thinking about visiting.)
The bus station is located adjacent to Den Haag Central, which is another great way of exploring The Hague. Bus drivers and tram operators will accept cash (in euros only) for payment.
Go ahead and get wandering about. The Netherlands is a beautiful country with lots to see! Enjoy your journey!