If you want to make your life easier, I would suggest AKL-EZE connecting to EZE-GIG. Flying all Aerolineas Argentinas, and probably staying in the same Int'l terminal would be better than having to change airlines and make many stops (Papeete, Easter Island and Santiago). I think 3 is better than 1. If you want to start getting your 'round the world fare, you may want to look into finding a cheap 1 way from AKL-EZE on either Qantas or Aerolineas. Or if those are all expensive buy a round trip and don't use the return. Then start using your Star Alliance package by flying on a Varig 737 EZE-GIG. Or you could fly all star alliance and connect in LAX on NZ or UA and then fly RG to GIG with a stop in GRU. Whatever you do, enjoy your trip.