Qantas/American is offering flights to Los Angeles from Christchurch for $739, Wellington $749 and $785 from Auckland in October & November & Jan/Feb. Also on sale are flights to San Francisco from Auckland for $796, Wellington $802 and Christchurch $840. These flights include bags, meals etc but they do connect via Sydney, Brisbane or Melbourne. However, AA is flying direct from Auckland for $785 - Google Flights $811. There are some other sale destinations in the US but nothing as cheap as LA.
Find the flights via Google Flights then use Skyscanner or whatever site to book your flights.
Auckland to LA $752 or $785 direct via Skyscanner - Google Flights $811 - 1 day stopover in Melbourne $926
Wellington to LA $ via Skyscanner $749 - Google Flights $808
Christchurch to LA $ via Skyscanner $739 - Google Flights $806
Auckland to San Francisco via Skyscanner $796 - Google Flights $869
Wellington to San Francisco via Skyscanner $802 - Google Flights $887
Christchurch to San Francisco via Skyscanner $840 - Google Flights $884
