Wellington is a great place to go tramping. A ton of day, weekend and longer trips are available within 4 hrs drive. Some popular places are the Tararuas, which were the birthplace of recreational tramping in New Zealand, the Rimutakas, the Ruahines, and Taranaki and Tongario further afield. Some popular trips in the Tararuas are the Northern, Middle and Southern Crossings; and Mt Holdsworth-Jumbo Loop; and Mt Holdsworth-Kaitoke.
Some links to look are:
Wellington Clubs: Tararua Tramping club www.ttc.org.nz and Wellington Tramping and Mountaineering Club www.wtmc.org.nz The clubs are a really good way to get a feel for tramping in New Zealand and the two I've mentioned are some of the largest and most active.
A really good website for trips and forums is www.tramper.co.nz.
DOC is the government body that looks after the conservation estate and as part of that they maintain tracks and the hut system. If you are going to do a few trips and using huts it is worth getting an annual hut pass rather than buying hut tickets (usually 1 or 2 per night) www.doc.govt.nz