Private Health Insurance for NZ Citizens

Hi everyone! So the time has come where I have seen myself setting up to transfer from NZ to Aus. I have managed to secure a job while still in NZ in Sydney and I will be moving there this January 2019--fresh start!

Now, I am wondering, as a NZ citizen, would it be wise to get a private health insurance over in Australia? (I will be living in Sydney). Also, can you give suggestions for a good, comprehensive health insurance in Australia? Provided if it's okay if we name drop company names here.

Furthermore, I still have life and private health insurance here in NZ. Do you advice for me to cancel it before I leave NZ or keep it going until I manage to secure a private health cover once Im in Sydney?

Your input will be much appreciated guys!