[Caveat: I am a Kiwi, and I ski a lot in NZ, but I have not skiied the South Island. Other Kiwi skiers will know more than I do about this topic, but this should get you going in any case.]
First things first, do not go to the North Island. There is skiing there, but it's not worth travelling that distance for.
In the South Island, there are two areas you should consider. The first is Queenstown
. It's towards the bottom of the South Island, on Lake Wakatipu. It's a "resort town", and the skifields are from 1/2 to 1 hour away. The skifields are The Remarkables, Coronet Peak, Treble Cone and Cardrona. As I hear it, Treble Cone is the most challenging.
Prices for everything in Queenstown, including accomodation, are pretty damn high these days. Accomodation can be hard to find if you're a "worker". Wanaka
is a town about an hour from Queenstown, and Treble Cone and Cardrona are often skiied from there...you'll probably find that prices are cheaper in Wanaka. It's a growing destination town, so undoubtedly has less night-life than Queenstown.
The other skifield you might consider is Mt Hutt. It's about half way up the South Island, and Methven is the town that services it. It used to be a pretty sleepy little town but I'd expect that's changed by now. It won't be like Queenstown but it'll probably have some of everything you'd need, although as with Wanaka, after-ski jobs will probably be harder to find.
My advice to you is to go to Queenstown/Wanaka. Mt Hutt is very exposed and often has to close due to the wind -- its nickname here is Mt Shutt. The skifields around Queenstown have more settled weather.
Our ski season usually starts around the middle of July and goes until the end of October. You will see earlier start dates mentioned -- such as the "official" start to the ski season, Queen's Birthday weekend, the first weekend in June -- and it can happen, but something between mid June and mid July is more likely.
There is some snowmaking at the southern skifields, but I don't know how much and it won't be as much as North Americans are used to. We are much more dependent on natural snow. OTOH, at least some of the southern skifields are tussock under the snow and so it doesn't take much to get the season underway. There is no tree skiing in New Zealand, to my knowledge.
You need to bear in mind that NZ is a tiny place -- only 3.5 million Kiwis in total, contained in a country that's about 2/3 the size of California, with about 70% of the population in the North Island. Skiing is a growing sport here, but you will find it quieter than you're used to, almost for sure, both on the slopes and in the nearby towns. There is a smaller market for everything one can spend money on here, just because of the population, and that has an effect on the types and quantity of services and goods on offer. This may not bother you, but it will bother some folks, so you ought to be aware of it. Also bear in mind that our dollar exchanges to the USD at around 50c typically: i.e. it takes two of our dollars to get one USD.
HTH and I hope you have a great time down under! Many Europeans and North Americans come here to instruct during our ski season, and many come back year after year.