New Zealand is full of excellent short tramps and many of them are well serviced by both huts and transport, meaning you can get away without a car or tent.
The great walks are justifiably named and the infrastructure makes them very convenient for travellers. They are, therefore, also more expensive and crowded. Locals tend to do them in the off season as it is cheaper and you donít need to book, so you can time your trip against weather windows, but although I often hear people disparage the crowds I never hear anyone complain about the quality of the walks.
There are many other walks that are almost as convenient but donít have the same volume of usage or the booking system. The Rees-Dart, Greenstone-Caples and Wilkin-Young are obvious examples (check for progress on the rebuild on Siberia Hut for the last one). From the Rees-Dart you can cross the Cascade Saddle into the Matukituki, although this is considerably more serious and makes the logistics a bit more awkward. The Matukituki itself is well worth a visit and has excellent possibilities, especially if you are happy to enter moderate alpine terrain.
The great walks, especially the Routeburn and Kepler, probably tend to be easier than some of these other options. The track from Wanaka to Arrowtown is reportedly straightforward with great views, but the logistics are a bit fiddly.
Around Mt Cook Village the trip up to Mueller Hut is superb as are the short trips up Sebastopol and Wakefield. If you hire some alpine kit from AGL you can access more diverse terrain.
If you donít have a car, atomic can move you around the island cheaply (also try the naked bus) and these other operators do good tramping logistics:
As people have commented above, trampers die in NZ primarily by trying to cross rivers (and side-streams!) when perhaps it would have been wiser to turn back or wait and also by slipping on wet snow grass / alpine tussock or snow with a bad run-out (think Cascade Saddle, Sefton Biv, Ball Pass), so take care.
Iím not aware of anyone dying from sand-flies but if you stop in the wrong spot it feels like it could happen. Most people happily leave tents behind; DOC will recommend a hilarious giant safety garbage bag.
You can get by in Wanaka using the shuttle to get out to the crags and then hitching back and you should be able to meet people from there, if you donít mind a bit of messing about.
If you just go sport climbing, you wonít miss out on much. There are good local guidebooks for Wanaka & Queenstown, which support bolting funds or you can get adequate coverage from the current version of the South Island Rock Guide. At Mt Cook you can get a topo from AGL for $1 or so.
Wanaka is fun, convenient and varied with lots of routes on a bunch of crags from mid-teens to mid-twenties and a few multi-pitch routes way up the valley. There is a pleasant day to be had at Kingston (great views) but Wye Creek is the best Queenstown cragging: lots of nice walls from 18-22 (although I thought it got quite same-same after a bit). The multi-pitch area up the South Branch of the Wye is well worth the visit. The approach is not as long as suggested and you can go in and climb 10 or so pitches (mostly around gr 20) with just a 60m rope and draws.
There is also lovely multi-pitch sport climbing (although a rack would give you more options) at Chinamanís Bluff past Glenorcy & some varied sub-alpine climbing - from snow and ridge scrambling to quite adventurish routes- in the Remarkables, but you will need some gear.
Long answer; I am rather excited by the thought of summer. Have fun.