Traveling tips: There are a number of airports in the area of Groningen which offer cheap flights to major cities in Europe

Bremen
There is a bus that goes a couple of times a day from Groningen train station to Bremen airport.
The trip takes about 2.5 hours.
You can find flights at: www.ryanair.com

Eindhoven
There are two trains an hour going from Groningen central station to Eindhoven central station. From these you can take a bus that drops you off at the airport.
From there you can also fly with ryanair: www.ryanair.com

Schiphol
This is the international airport in Amsterdam. There are two trains an hour going from Groningen to the airport, one of them goes straight, if you take the other one, you have to change in Amersfoort, where you will get a connection 2 minutes after you arrive.
Other Destinations:

Berlin
A popular city to go to from Groningen is Berlin. It’s recommended that you rent a car (www.hertz.com/nl) to go there. It will take you about six hours to get to Berlin. It is a huge city, so taking that into account along with the total driving time, you should really consider going for a weekend at the least.

Islands in the North
The Netherlands has five populated islands in the North. The two islands to the most-right are Ameland and Schiermonnikoog. These two islands are easily and quickly accessible from Groningen.
The easiest way to go is to buy a bus-boat-busticket at the Arriva-office at the station. This ticket includes the busride to the harbor, the ferry to the island and also allows you to take the bus on the islands (On Schiermonnikoog there are now cars allowed, only taxi’s and buses). Of course the return trip (on the same day) is also included.

If you prefer to stay for a weekend, that is of course possible. However, the ticket is only valid on the day that you travelled to the island. So you will have to buy separate tickets to get back to the city. In case you come from Ameland and take the ferry back to Holwerd, you can take bus 134, which will take you to Groningen. From Schiermonnikoog you get on the ferry back to Lauwersoog where you can get on bus 163 which will also bring you to Groningen.
In case of doubts on how to get to the islands or back, you can always ask one of the service people / bus drivers. Most of them speak English and are more than willing to help you.