The waag (weighing-house) dates from 1595 and was built in Manneristic style. There are similar buildings in several other towns in Friesland. You can get a drink here, and perhaps something to eat.
Jugendstil or Art Nouveau buildings in the Netherlands usually are in quite moderate variants, but this pharmacy from 1905 is a bit more flamboyant than usual.
The Jacobijnekerk or Grote Kerk (Big Church)is one of the oldest churches in Leeuwarden. It is very centrally located near the Fries Natuurmuseum.
Many people forget that leeuwarden is also a big student city, just try something from Leeuwarden nightlife.
Go and see the staion and the centre of town. Furthermore pay a visit to the airbase when there is Open House (this year that was at july 6 and 7)!