Flatting in Auckland can cost you about max $100 - $180 per week but I'm sure you can share a place with someone for much much less than that especially if you are willing to commute. My cousin is renting a 1 bedroom studio and it costs him around $180 and this is in Auckland which is one of the most expensive cities in NZ. If you are in a small town you can probably rent a whole house for around $130 per week not including any of the bills.
You will need to consider how much it costs to travel to where you work with how much the room costs and whether it includes all bills like water, electricity/gas, internet etc because the buses in Auckland are quite expensive (or I thought so anyway). So it might be an idea to consider living close to where you work. Biking in Auckland is an affair because there's a lot of hills so unless you're very fit, I wouldn't recommend it.
Walking in Auckland is good but only if you're going short distances because things are quite spread out.
Good luck.