aryaya Welcome to FT (and possibly Aussie)

As above. you have the option of using Ultimate Rewards points by redeeming them for a paid ticket at 1.25 cents/point.
Or move the UR points to a frequent flyer program and try for an award flight. Awards flights are not free nor guaranteed to be available. Award availability limited. All awards have taxes and some have cash surcharges.

Awards flights USA to Australia can be hard to get. They become available 355/330 days out. Awards for 3 plus a lap child on the same flight will be much harder to get. Being realistic you may get 1 or 2 awards and buy the other/others with cash. Most airline require you to pay 10% of the full fare for a lap child. Depending on the airline this can be many $$$$.

Airlines that fly USA/Canada to/from Australia (not from all airports to all airports)
- Air Canada AC (YYR – SYD, BNE from Jun 2016) [Star Alliance]
- Air New Zealand NZ (LAX SFO IAH YVR via AKL) [Star Alliance]
- Air Tahiti Nui TN (LAX via Tahiti. Last segments on codeshares) (AA & DL partner)
- American Airlines AA (LAX - SYD) [OneWorld Alliance] (AS partner)
- Delta DL (LAX – SYD) (VA & AS partner) [Skyteam Alliance]
- Fiji Airways FJ (LAX SFO HNL via Fiji) (QF, AA & AS partner)
- Hawaiian HA (HNL – SYD BNE via Hawaii) (AA & VA partner)
- Jetstar JQ (HNL – SYD BNE)
- Qantas QF (LAX SFO JFK DFW HNL YVR*– SYD BNE MEL) [OneWorld Alliance] (AS partner) {* seasonal}
- United UA (LAX SFO – SYD MEL) [Star Alliance]
- Virgin Australia VA (LAX – SYD BNE) (DL partner)
And others via Asia & Middle East

Some more information here http://www.flyertalk.com/forum/infor...help-here.html