I would suggest settling down in the city for a few days/a week or so and deciding where you would like to live for the longer term. The thing is for most people when they choose a rental; budget may not always be the main consideration. A cheaper place may not always be a good decent rental or may be a long way off from bus routes or may have terms and conditions of rental that are hard to meet. If you narrow your search from very early on, you might find that suburb or place to be less than desirable for you. For example, Onehunga looks good on paper but in reality it is quite a distance from the city centre and taking a bus would probably take longer than you realise. Furthermore the suburb is rather industrial/commercial in nature and though there are residential houses, it isn't as attractive as other parts of the city. If I were you I would put my bet on Mt Roskill, Mt Albert but they are usually more expensive to rent. When you land, take a little time to explore the city and decide where you would like to stay. It is better off than signing a tenancy agreement and paying off the bond only to find yourself breaking the contract because you can't bear to be in the place or neighbourhood.