Hey there. I live in Dunedin.
Sounds like you have a great hoilday planned.
Firstly, you asked about getting a 4wd rental car. I would say no. Our winter is June, July, August so you wont actually be here for the 'extreme' snow. You will definitly need chains though. On some of the roads up to the ski fields it is nessary and you must present your chains to the officers up there. I'd suggest you ask the rental company if they supply chains. If not, then you can hire them from recreational shops like R and R sport or from Otago University Rec hire.
I'd also strongly suggest you have an itinerary before you leave. You don't have to follow it completely but it would be extremely helpful.
Places I've visited that I would recommend would be:
Christchurch
Hamner Springs
Nelson
Mt Cook ( Glaciers!!!)
Tekapo
Wanaka
Queenstown
Manapouri
Te Anau
And the list goes on.... Seriously! Find a good website such as
www.newzealand.com/travel and find the main attractions in the places you want to visit. Also try and budget it.
I've never been to Mt Hutt but I've been to Wanaka. It's pretty expensive there though.
I suggest also that you find out a bit more about the ski fields around this area because some fields such as the Remarkables are designed for more advanced skiers.
Also look for good deals for skiing. I don't think it should be too expensive for a day skiing. Try
www.cardrona.com I think that's a begins field.
Goodluck for your hoilday. If you've got anymore questions I'd be happy to answer them!