Maastricht has three train stations: Maastricht Noord in Limmel/Nazareth, Maastricht Central station in Wyck/city centre and Maastricht Randwyck, in Heer/Randwyck. Trains leave from Maastricht Randwyck to Maastricht Noord via Maastricht Central station at least twice an hour – same goes for the other way. Maastricht Central Station has the most international connections to Belgium (Liège) and national connections to all the parts of the country including Amsterdam and Schiphol (Amsterdam Airport).
If you want to travel within the Maas-Rhine Region you can take one of the Arriva trains. Arriva is providing regional public transport within the whole Euregio Maas-Rhine, which expands well into Germany and Belgium. You can find a map of the area covered by Arriva here.
If you want to board a train to travel longer distances, for example if you decide to go to Amsterdam, you will have to take one of the trains provided by the national railroad company NS.