Hi Elderberry
I'm an Aussie living in Auckland in the north of the North Island and have been here for 5 1/2 years. My DD was born in October 2000. October-December can still be quite cool, even in the north of the North Island. Because it's very long north-south, you'll need clothing for all seasons if you're planning to travel both islands as it can be 20 degrees celsius in the north and 8 at the bottom of the South Island! But saying that, some places in the south have 'heatwaves' where it can be 30 degrees one day and 15 the next.
You can hire equipment short term like car seats and strollers (maybe even portacots) from several places but you'll have to be quick - I can get you contact numbers if you need them. No use lugging it on the plane if you're able to get it here.
Looking at Suew's message, nappies here do have velcro tabs - both Huggies and Treasures Ultra - if you're after disposables. Parents Centre is a good site, too, and have a discussion board. Also try www.newmother.org.nz as they have a discussion board too where you could post a message about travelling in NZ.
If you're breastfeeding, doing so in public is not a problem as NZ is pro-breastfeeding and mothers are really encouraged to do it. If you need to take your son to the doctor, health care is very accessible but being a non-resident you'll probably have to pay (health care is free for under 6yrs here).
Will you be visiting family, north/south island? There's plenty to see and do, scenery is magnificent. Heaps of adventure tourism - bungy jumping, white water rafting etc. Also the opposite with plenty of relaxing things to do like walks, hot pools, museums, art galleries, etc. If you'll be driving a lot, I find the roads outside the cities are sometimes not that great.
If you have any specific questions or want any phone numbers re hiring equipment I'd be happy to help.