There are three train stations in Budapest: Déli, Keleti and Nyugati.
Trains to Tokaj leave from Keleti or from Nyugati. Make sure to choose a train that is marked IC intercity.
To travel from one train station to another the easiest way is to take the metro. You can buy a ticket for 350 HUF at a newspaper stand or from the purple vending machines. Even if you travel on 2 different metro lines, you can use the same ticket provided that you change underground.
There are four metro lines in Budapest, each one with a different colour. Click here for the Metro Railway Network map of Budapest.
For detailed information about public transport in Budapest, please click here or here.