Aside from the rates, there is the monthly cost of paying for water (some charges are fixed), and of course property maintenance on your house which is your own doing.
As pointed out, rates are tied to the government valuation of your property, the region and suburb you live in etc but generally you can expect to pay around $2000 per year or more for a medium, average house in Auckland.
Many banks pay off the legal fees when you obtain a loan. However you need to be a resident of NZ or at a minimum, work visa holder with a full employment contract in order to purchase a house in NZ with a bank loan.