2
You've got only a few options.Qantas fly from Sydney to Santiago, LAN i think still flies from Sydney and/or Auckland to Santiago and Aerolineas flies from Sydney to Buenos Aires.
Otherwise fly to LAX and back down again. Used to be able to fly via Tahiti and Easter island, but no longer able to fly Tahiti to easter island.
3
Aerolineas Argentinas is probably the cheapest, but it will be a horrid flight (poor quality vampire movie on one screen at front of plane; the fight attendants disappeared for most of the flight; no one at check in counter till about an hour before the flight) . Pay the extra and go with a different airline.4
Probably easiest (and maybe cheapest depending on current offers) to fly with LAN direct from Auckland to Santiago de Chile.That's what I'm doing in August. Flights were cheap as!
5
LAN Auckland to Santiago was a lovely experience. And that was on Christmas Eve last year.7
Even the journey from Auckland to the airport was cool, thanks to, I think, Westwood. We skipped getting the airport express bus, always hate them, a rip off for something you can do cheaper yourself, and got a proper train from the Britomart out to Papatoetoe and the bus from there. Had a bit of a nightmare at the airport because someone at Oneworld mucked up a change in our flights but a nice young Chilean lad sorted it out whilst wearing a father christmas hat. The plane was lovely (Always wanted to go on an A340), food was alright, free Chrimbo champagne, decent selection of films and we flew over the Corromandel where we had spent the previous month and a half. Had a cracking view of Thames and Waiomu.8
I'm flying with LAN from Auckland to Santiago next week and when I come back i'm flying with LAN from Santiago - Easter Island & Easter Island - Papeete, Tahiti and with Air Tahiti Nui from Papeete to Auckland.I believe the AKL - SCL service is daily but the Easter Island - Papeete is only about once a week.
If you want to see Easter Island and travelling to/from NZ - going through it works out to be much cheaper than return flights from NZ to South America and return flights from mainland South America to the island.
Good luck!

