Side trip from Maastricht

Aachen, home of Charlemagne and his cathedral, spa water, lovely small city. Very near.

Nigel - you are just a little bit quicker :) , indeed Aachen (with the bus 1h)

From Maastricht you can visit:

Koeln needs 2½ to 3 hours with the train.
Monschau in Germany, very touristic but really nice, 2½ hours with the bus via Aachen / 2 changes.
Thorn, cute village, train and bus 1:10h.
Castle Alden Biesen, near Bilzen in Belgium west of Maastricht, train to Bilzen 1:20h and bus or taxi.
Tongeren for Gallo-Roman museum, 1:10h with train.

You can make a touristic tour through the hilly countryside of Zuid-Limburg and even cross the Belgian or German border, a car remains the best way to do that. Valkenburg is touristy, but worth a stop. The Voerstreek just south of Zuid-Limburg has nice spots to see or visit like Abbeye du Val Dieu and castle Commenderie. Ask your hotelkeeper for more tips and further detailed info, they know best.

For planning public transport in the Netherlands use: http://9292.nl/en and for international trains for instance: https://www.nsinternational.nl/en

Along with Aachen come Dusseldorf and Cologne, major cities that can't be seen in just one day. Cologne and Aachen are particularly well connected to Brussels. If your train goes through Liege, be sure to stand on the carriage steps during the stop, to see the train station by Calatrava.

Without going into details, it might be wise to consider the many, many attractive cities that are within one hour of Brussels and Amsterdam, instead of treating Maastricht as a hub. Depends on your schedule.

Oh I see my mistake: Koeln or Köln is German for Cologne.

Hassel isn't a city that most people would (or should) go out of their way to visit, but if you're in Maastricht for a little while, give it consideration. It's pleasent enough, if not particularly unique. If you have any interest in Jenever, the city has a museum devoted to it. And for some reason, this little city is known for it's high-end fashion boutiques. Even if you don't buy anything, the fashion choices of the local population illustrate this.

Open air museum Bokrijk east of Hasselt is really nice too. 2 hours and 1 change with the train or final leg with the bus.

Thanks, Wil, I forgot about Bokrijk. Of all the open-air folk museums in Europe, I think it's one of the best.

There is a World War II American Cemetery about 6 miles out of Maastricht called Margraten the Netherlands American Cemetery and Memorial. When I visited it, I was surprised at how moving it was. I was not expecting that. It is very worthwhile since you are in the area. Also, I vote for Monschau, Germany if your going for a side trip a little farther out. Yes, it's very touristy but that's because its so beautiful.

Thanks too Tom, if you didn’t mention Hasselt I wouldn’t have thought about Bokrijk too.

The Castel Gardens of Arcen can be of interest too. Train to Venlo and further with the bus, needing some 2 hours. Can in combination with a visit to well known brewery Hertog Jan, same bus line but further north just outside Arcen.