Best idea is to do a search on the internet. Flights are available from Auckland to Bali but it is a matter of checking various airlines to see who is the cheapest. Also look at airfare and accommodation packages as it may be more economical than just airfare.
Garuda fly from Auckland but not sure which other airlines offer flights.
You might like to check the website for Zuji which looks for the best prices. I have used them before and was very happy. From New Zealand there is always a stop over in either Australia or if flying Singapore airlines the stopover is Singapore. Either way it's about 21 hours travel time.