Denise,
Above 3000 metres in New Zealand there are opportunities for mountaineering but not trekking. These opportunities lie within Mt Cook National Park or the summit of Mt Aspiring. For mountaineering trips Adventure Consultants is a well known operator.
If you are interested in trekking, then to get above 1000-1200 metres in New Zealand will get you above the bushline. There are many opportunites, mostly in the South Island with a few opprtunities in the mid-lower North Island.
Send me an e-mail if you want further info.
Grant Newton
Chief Guide – Wellington Tramping & Mountaineering Club