This is what I'd do: book in at Base
http://www.basebackpackers.com/wellington.htm backpackers & get a private double room for two or three nights ( about $80 per night if memory serves). The rooms they have are excellent quality ( banish any thoughts of crowded dorms & traveller's feet - these are gorgeous & as central as you could wish for)...as good as any 3 star hotel...and then look in Wednesday's & Saturday's Dom Post, on the trade-me & Finda rental websites for a place to stay.
If you're looking to live in central Welly ( ie. the immediate burbs) you'll be well-advised to head for somewhere that gets afternoon sun. Very important ( you'll see why when Winter hits town)
Try the Eastern suburbs for proximity to town tempered by more affordablity & peace & quiet ( Hataitai, Roseneath, Miramar, Kilburnie). Mount Victoria, Newtown and parts of Mount Cook are nice too ( and closer to town), but can be more expensive & noisier.
Any other advice, just PM.