We used the TOP 10 motel/camp sites when we first got here.
The Northshore Top 10 in Auckland has some really nce self-contained units for $110 per night. This soon mounts up. There are cheaper units if you use the communal facilites and bring your own bedding.
We are currently living in a very nice fully furnished 1 bed granny flat in Birkenhead, Auckland. This costs $300 per week and includes all power and water. Not too bad at all really
Also this is a short term rolling lease - we can give 3 weeks notice to vacate.
Whoever said that fully furnished accommodation was hard to find was right. It is out there but you have to be willing to put in some legwork and view some real pits. Usually, if there's a fully furnished apartment for rent in the city centre and it seems really cheap it's probably because it's a hovel, or perhaps we were just unlucky
Also be advised that for city centre apartments there is usually an extra weekly rent for a car park.