I'm in Auckland. In terms of choosing a suburb, I'd really recommend staying as central as you can. Any of the city fringe suburbs are fantastic, but can be pretty pricey. Lots of Brits end up over on the north shore, either Takapuna or devonport. Great to have beaches right there, but the bridge is a nightmare for commuting unless you travel off peak hours.
Public transport is pretty dismal, so you really will need a car - one each unless you end up in Ponsonby or something.
It's uncommon to find a furnished rental, so if you can ship out basic furniture that would be ideal. Have a look at www.freedomfurniture.co.nz to get an idea of local mid range prices.
www.trademe.co.nz is our version of eBay so look on there for rentals as well. I know a few people who have moved over and have lived in apartment complexes to start with. The Beaumont is pretty popular - unfurnished, but whitewater included. Complex has a pool and gym, fab supermarket and park adjacent. Also only 10 mins walk to the waterfront and loads of cafes / restaurants.
Re childcare, it's easy to get a place. Have look on www.ero.co.nz - that's our version of ofsted, and you can read reports from all schools and daycares. No such thing as 18 month waiting lists - well I've never heard of any!
Plunket will definite be able to point you in the right direction. Play center is good, but I think they expect a lot of parental contribution. Still, good way to make some friends I'm sure!
Best of luck with your planning. Feel free to ask me any questions and I'll do my best to answer.