I think I can answer some of your questions, I am living in Auckland.
Speedway here is dominated by car racing which unfortunately means heavy and sticky clay track surfaces for the bikes. This means that unless you are watching at a bike only track you are unlikely to see much good bike racing. There are a huge number of tracks here but most of them don't run bikes. Auckland, Invercargill, Christchurch and Napier all have good bike tracks.
There is no league racing as such. There have been some good attempts at getting a league started but costs have been prohibitive.
You will get to see the GP's eventually on SKY. The first ones will not be shown here until well into the European season. Some people do have them taped and sent over so they may be available that way with a contribution towards postage perhaps.
Do you know where you will be living here? I can then let you know the tracks nearest to you although a lot of them have finished their fixtures for this year. I can then also give you some website addresses to have a look.