Leeuwarden is a city in the Netherlands. It is the capital of the Dutch province of Friesland. Hitching out is pretty straightforward.
South towards Utrecht
Go to the central bus station and take bus #94, #95 or #28. Get off at the stop at the end of Hendrick Algraweg (the beginning of the Rijksweg if the bus driver doesn't know). Continue walking along the main road (A32) until the Esso garage - you should be able to see it from the bus stop.
This distance is also walkable from the centre. Just follow directions for the A32 south.
You can also use this spot for the direction of Groningen, but it's less common and you should therefore consider making a sign to prevent cars going south to stop for you.
Go to the bus lane on the Drachtsterweg just past the Drachtersterbrug (the bridge over the water). This is one of the main roads leading out of Leeuwarden towards Groningen and Drachten. Go stand where the bus lane starts as cars have a good chance to stop there for you. The bus stop further down the road is less than ideal since cars for Groningen are in the middle lane and can't easily turn over to the right anymore. To get to this spot you can walk or take a bus (14/28/94/95) to Drachten and get out at that bus stop (Drachtsterweg).