Check Trademe or websites of agents directly e.g. Barfoot & thomson, Ray&White etc.
Try looking for properties within this circle -
Point Chevalier, Grey Lynn, Mount Eden, Epsom, Greenlane, Onehunga, Meadowbank, Glen Innes, Orakei, Glendowie, St Heliers, Davenport, Takapuna and Birkenhead
All these areas will need relatively less and direct transport than changing 2 buses/trains.
Some of the areas have buses travelling through motorway during peak hours which cuts down travelling time by 30 minutes (city rush), if possible do keep an eye on this.
Are you exploring to stay as PG to start with? I had one of my colleague who moved with his wife (no kids) and PG accomodation worked out quite well with them especially getting used to localities, distance, familiarities with Auckland city, look for job, time to save deposit, buy car & household stuff etc. Not sure if this is some option you are thinking of. All the best