Wellington is the capital of New Zealand.
When arriving in the Wellington area, you might be able to catch a local train, but even at night it's worth just trying to hitchhike into the city if you're north of the point where the motorway starts. If you are dropped off along the highway, you can try hitching from an onramp or you could take the train into the city.
North up SH1 towards Bulls/Taupo/Taranaki/Hamilton/Auckland
Catch the train to Paekakariki (Kapiti line), walk out of the station towards the direction the train arrived from. Turn left onto the highway, there is a good spot with a bus stop bay just 20m further.
At the corner of Taranaki St. and Arthur St., close to the top of Cuba St. - there is a little space, but not very convenient for cars to pull over here. Still might be worth a go if you don't fancy walking.
From the main Wellington train station, walk along the main road (Waterloo Quay) towards the stadium. Follow the road as it bends around past the stadium (the street name changes to Aotea Quay). Follow this road until it crosses the motorway (SH1), cross under the bridge and you'll find a bus stop on the right. Shouldn't take more than 20 minutes to get a ride, just try not to accidentally flag down a bus.
North up SH2 towards Masterton/Hastings/Napier/Gisborne
Take the train to Upper Hutt then the number 110 bus to "Maoribank, Mangaroa road near SH2". Hop off the bus and cross SH2, there is a great spot where the traffic lights force the traffic to stop and there is a pull in area - on google maps it is 1183 River Road Clouston Park, Upper Hutt. We got picked up in 5 mins.
Ferry to South Island, Picton
You pay (normally around $45-55) per person. And it's better to book in advance although be aware of deals around the start and end of the university year, when specials sometimes crop up.
If you ask around a bit on the ferry it will be easy to get a ride to Nelson or other places in South Island.