Wellington is the capital of New Zealand.
When arriving to Wellington, you might be able to catch a local train, but even at night it's worth to just try to hitchhike into the city when you're North from the point where the highway (motorway) starts. If you are dropped off along the highway your can try hitching from an on ramp or you could take the train into the city.
A common approach to getting out of Wellington on Route SH1 to the north is from Paraparaumu. "Paraparam" is far enough out that one can avoid all the short trip cars going to work from Wellington. To get here take the train from Wellington station which costs about $6.50 (ticket inspectors are always on the train) and takes 55 minutes. At Paraparaumu station, exit the train and turn right to walk north to the end of the platform. Continue this direction to the road and turn left. Turn right at the lights and this is Route SH1. You can hitch here but it is very busy, so continue about 800m further north along the road, around the curve till you come to the nice big lay-by. The perfect hitching spot! It may seem like alot of time to get out here, but in reality you have shaved this time off your total trip since it takes almost an hour to get here by car from Wellington, especially if the traffic is bad.
Ferry to South Island, Picton
You can't take the ferry for free. Guaka hasn't tried hitching a ride from cars going on the ferry. 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 though.