First you will have to determine your budget and what kind of house you need/want. Rent here is paid by the week and housing is quite expensive. Before you get to move into a new home, you will probably also have to pay 2 weeks of rent in advance and a bond (maximum of 4 weeks rent).
Finding houses can be done online on the property section of Trademe.co.nz or their app specifically for property. There are loads of listings here and we almost exclusively used this website to find our home. Another option is to check Facebook groups for people in New Zealand, we joined a Facebook group for Dutch people here and got in touch with someone renting out their place through that. Or if you already know some people here, you could ask around if they know any places looking for new tenants.