Some info on flights.

US AA & BA are all OneWorld as is Qantas
UA is in Star Alliance as in Air NZ (but hard to get Air NZ award flights)
Flights Australia - NZ are relatively cheap and from CNS, BNE, OOL, SYD, MEL & ADL to AKL, WLG, CHC & ZQN (but not all combinations)
By Sept 2015 your US & AA miles may be combined as AA miles (no announcements yet)

Airlines that fly Australia – New Zealand are
- Air New Zealand (codeshares with VA) [Star Alliance]
- Virgin Australia (codeshares with Air NZ) (Delta partner)
- Jetstar (a low cost, low quality carrier with low-medium-high priced fares owned by QF)
- Qantas (codeshares with EK) [OneWorld]
- Emirates (codeshares with QF)
- LAN (SYD AKL) [OneWorld]
- China Airlines (BNE/SYD to AKL & SYD to CHC) [Skyteam]

Airlines that fly in Australia domestically
- Virgin Australia (Delta partner)
- Jetstar (a low cost, low quality carrier with low-medium-high priced fares owned by QF)
- Qantas [OneWorld]
- Tiger Airways (a low cost, low quality carrier with low fares part owned by Virgin Australia)
- Regional Express Airways (REX) Regional airline

Airlines that fly in New Zealand domestically
- Air New Zealand (Star Alliance)
- Jetstar (a low cost, low quality carrier with medium priced fares owned by QF)

Airlines that fly USA/Canada to Australia (most from LAX) are
- United [Star Alliance]
- Qantas [OneWorld Alliance] (AS partner)
- Delta [Skyteam Alliance] (VA & AS partner)
- Virgin Australia (DL partner)
- Hawaiian, via Hawaii (AA DL & VA partner)
- Air NZ, via AKL [Star Alliance]
- Fiji Airways, via Fiji (AA & AS partner)
- Air Tahiti Nui (last segments on codeshares) (AA & DL partner)
- Air Canada [Star Alliance]
And others via Asia & Middle East