You'll have to get around within Australia somehow, since you want to visit far-flung parts of this very big country.

From/to the US on one side, go to/from the biggest city you're visiting in Australia (that would be MLB on your list). From/to the US on the other side, go to/from New Zealand or wherever non-Australia you're visiting. Then all you have left is intra-Australia and Australia-New Zealand, not the more difficult routing direct between a relatively smaller city in Australia and the US.

Then figure out if it makes sense to use separate premium awards or not for the shorter connections within Australia and within New Zealand and between Australia and New Zealand (or wherever). It may be much easier for 4 to find paid flights than award flights, so where you can afford to do paid flights for shorter legs that may make much more sense.

But, btw, also consider how much total time you have. Just getting from one part of Australia to another will take significant time, and then you want to allow enough time to see whatever there. And New Zealand is somewhat far-flung too (it's a narrow but long country). So make sure that you really have enough time for both countries (because it much simplifies the routing if you're only going to one or the other, rather than trying to throw a bunch of stuff in just to say you've done it).

It can already be hard enough to find awards for 4 to anywhere. The more complicated those awards have to be (because you're flying roundabout to stop, not just flying roundabout to get 2 other people there) the harder. You may actually have to split the family up between two different routings just to get 4 seats to the final destination!