In reply to skirton:
I've done a fair bit in NZ over the years. I live in Sydney, so it's the nearest alpine area to me.
Probably the best areas would be from the Mueller hut above the Mueller Glacier and from the Tasman Saddle/Kelman Huts on the Upper Tasman Glacier. Both offer plenty of F to PD+ mountains within easy reach once you have got to the hut. These include Mts Kitchener, Annette, Sealy, Jean, Marie, and Darby (Mueller) and Mts Abell, Annan, Aylmer, Hochstetter Dome, and Darwin (UTG). The De la Beche Hut was demolished last year but you can still bivy on the site if you want to do classic easy climbs from lower down the Tasman such as Mt De la Beche and the Minarets.
You can walk up to the Mueller Hut in about 3 to 4 Hours from Mt Cook Village. Access to the Tasman Saddle/Kelman huts (they are close together) is best done by helicopter as the glacier walk up with a week of supplies and gear is something of an epic.
The best bet to find someone to climb with is to join the NZAC. That will also give you some priority in hut matters. Rescue is free in NZ. I know some of the guides at Adventure Consultants and can recommend them. However no guide/course is cheap!
The NZ Alps are quite low compared to the European Alps but they are subject to much worse weather that sweeps in suddenly from the west. They are lots of fun as there are usually few people about but you should not underestimate them.
Have a great time!