NZ has some great climbing and mountaineering. I lived there for 2 years so kinda know a bit but not all.
If your in the North island good rock areas are whanganui bay (at Taupo), Wharapapa South (Smiths Rocks, Froggatt) and Waipapa. In general the areas north of Auckland have difficult access or are very short. Whanganui absolutely rules and has some great trad and sport routes.
Mountaineering and Ice in the North can be found on taranaki and ruapehu. Good waterfall can be found at turoa ski resort but this is pretty much plain ice with a snow top out so good snow & ice protection is needed. Lots of good mountaineering routes up ruapehu. You can climb up to the summit and ski off the top, its pretty steep though so be warned.
South Island is excellent - Paines ford, Castle Hill, Port Hills above Christchurch, Long Beach and Wye Creek are awesome rock climbing areas.
Good ice and mountaineering at Mt Cook village as well as some really good big wall rock stuff. Pretty alpine rock (freeze thaw effect on rock causes shattering) so be careful. Easy stuff at sebastapol bluffs but it has up to 10 pitch routes. Twin steams is great back country climbing but go prepared. A nice easy alpine route for 2 - 3 days (difficult route finding though so take a map and compass and know how to use it - properly!!) is the ball pass on mount cook. If your experienced enough Mt Cook is great but well scary and dangerous. 4 people died in a serac fall 2 days before we climbed it so consider taking a guide or go with people who know what they are doing!!!!!!!!!
I never went but heard that the west coast glacier frans joseph has good ice climbing on it. Maybe check it out.
Check out www.climb.co.nz