North Island:
Auckland - Mt Eden Quarry has heaps of climbs at all grades.
Waikato - Many small crags dotted about the landscape, some local knowledge is neccessary and the topos are almost essential for the locations.
Taupo - Western side of lake, nice crag lots of natural lines around the mid grade and lower mark. Awesome atmosphere.
Wellington - Lots of bouldering at a couple of crags. Baring Head is good solid bouldering country. Titahi Bay is chossy but exciting.
South Island:
Paynes Ford and Pohara beach in the NW of south Island has an awesome sports climbing venue, check it out with the locals. Pretty close to some well known Hiking tracks - they call it tramping. Abel Tasman track and the Heaphy Track.
Canterbury - vast quantities of good rock and some not so good. Castle Hill is popular but you have to pretty tough to handle the snow in their winter.
Around Christchurch on the hills there are some awesome crags - get the guide book.
Further south there is plenty of bigger rock inland in southern Canterbury. Mt Somers and Cloudy Peak
In Fiordland, home to the famous Milford Track there is the big world of Big Wall climbing and some awesome summer alpine rock routes as well as awesome mountaineering opportunities. Tutuko, Madeline, Sheerdown, Pembroke plus many more.
Plenty of camping in most spots. Some crags can be camped at. Some are close to town. Renting a car and hostel accommodation in New Zealand should be all you need.
You can buy topos from:
In Auckland, Wellington, Christchurch go to Bivouac.
Palmerston North go to Mountain Equipment.
Christchurch also has McEwings and Mainland camping.
Equipment
Take a fullish natural rack with doubles of small wires. Lots of naked head bolts around the need to be slung with a wire with the nut slid up like on a neck tie. 8 draws will get you up most of the clip ups but bring some extenders just to cover the longer ones.
Climbing Gyms:
Auckland - Rocknasium
Hamilton - has a new gym go to Wildside (shop) they will give you the address.
Palmerston North - City Rock on Grey st and Massey Uni Rec centre have a small bouldering wall.
Wellington - Victoria Uni have a wall and there is a gym club wall cant remember the name.
Christchurch - YMCA have a wall and bouldering room
There are many more in the provincial centres.