In reply to greylag:
Picton to Chch is not far, so assuming you don't want to drive past Chch then come back up, it would make sense to go down the West Coast, which means you'll see the glaciers etc anyway. They may not be much interest to climbers to just walk up and look at, but if you can afford a glacier flight, or better yet get a helo up to one of the huts and do a climb or just hang out for 24hrs I think this is a worthwhile experience. The coastal scenery is also good, if you like that sort of thing. Arthurs Pass is en route across the mountains going west (unless you cross much further north) so you can do a day walk from there too (Avalanche Pk etc).
Further south past Fox you can also do an overnight walk in to Welcome Flat hut and sample the hot springs (smelly). The longer famous walks (Milford etc) are $$ and booked up in high summer but there are countless other things to do. An overnight at Brewster Hut, near Haast Pass, is also a nice trip if you want some exercise. Three weeks is not long enough if you want to actually do a number of outdoor things as the weather will shut you down 30% of the time.
Some touristy things climbers often disdain can be great. I did a half day of sea kayaking on Milford Sound in perfect weather that was like something out of a holiday brochure, and not expensive. Queenstown is branded as too touristy but it's a stunning location, fun activities and good cafes, restaurants etc. There are a couple of day walks out of Mt Cook Village that might be good in nice weather, and good to go there and at least get a good look at it. Plenty of air charters if you can afford it for sightseeing etc (weather dependent though).
Annoying little midge like flies in summer down the west coast, OK in winter, though cold.
Chch is a bit of a mess still, at least in the city centre. Akaroa is a nice day trip, French-heritage seaside village place, nice scenery etc.