Was just on a trip to Australia and New Zealand. These are details I have noticed, please update wiki as I am new, can not edit it:
1) In Australia, the preferred/first network it would connect to would be Telstra. Telstra everything works except MMS. I tried changing all APN settings, but MMS send/receive will not work.
2) In Australia, if attempting to switch to Vodafone AU(registration), it will take many tries (at least on my nexus 6p), and it will connect with MMS working well. However, I randomly would get Emergency Calls only, and will lose data access. Best bet is to use Telstra without MMS

3) In new Zealand, Vodafone NZ seems to be the best provider. I receive LTE most of the time.

4) If vodafone is unavailable in New Zealand (in some rural areas), Spark NZ works well. Registration might take a couple tries (often 3 for me, if not wait a little then try again). Spark also gets LTE.

On the nexus 6p, I find the search for networks times out often and I have to retry. I find that disabling mobile data temporarily then airplain mode on/off allows the network list again.