Remuera, eh? You'll be about 15 minutes away from my house
Edit: I can probably answer one or two of your questions, too.
You should find the cost of stuff to be reasonable. Pounds to NZ dollars always favours the pounds - we tend to earn less, but our stuff is cheaper
A few quick comparisons off the top of my head - A Big Mac Combo is around $6 (£2), a coffee would be $2-3 (70p - £1), one stage on a bus is about $1 (30p). A movie ticket is $13 (£4.30)
Transport around Auckland is pretty much limited to the bus or cars. There are trains but I have no idea how to use them. You can also get ferries easily if you're travelling over water.
Book transport well ahead, if you can. The ferry between the North and South island (the Interislander) is usually booked up months in advance.
Have fun here!
