Are you meaning under NZ$300 per week? For two of you? Anywhere near to town will be a shoebox for that price. You could flat with other people though for that price range. If you mean US$300, then you have a lot more options. Personally, I would go as close in as possible, even the CBD area. You'll save $$$ and time on commuting, and the $$$ can go towards the rent.
I had a 80sm warehouse / loft type conversion in High Street (plonk in the middle of the cool part of CBD) for $350pw if that helps. I think I got a good deal though, most inner city apartments are smaller and intended for students. look at rentals on trademe.co.nz and you'll get the picture.
CBD carpaking is anywhere from $10 to $18 per day. Monthly carparks are from $200 per month. Auckland is not a "public transport" city unfortunately, because of its difficult geography. Yes, there are plenty of buses and some trains, but it all takes longer and probably costs more than in other countries.
Honestly, I would just get a CBD apartment on monthly tenancy until you get your bearings. Unless you sign a fixed term lease (uncommon in NZ for residential) you can terminate on 3 weeks notice, so no big dramas. Bonds have to be deposited with Government agency, so usually no difficulty getting them back (just take some photos when you move in). An apartment is more likely to be partly furnished and have all the whiteware.