Does the OP want coach award tickets from a FF program or something else?

As others have suggested, it's helpful to have one account in each alliance so that miles are never lost, but you should also try hard to concentrate miles in one program to the extent that you reasonably can do so. This means focusing on one alliance and trying to book most of your flights on carriers in that alliance. Watch out for rules about miles expiring. Once miles have been credited to a program, this cannot be changed: you CANNOT transfer miles between FF programs, although US and AA miles are likely to be combined as part of the merger of those airlines.

Alaska has a good FF program that allows you to credit flights from different alliances, including AA, DL, CX, etc. This could work well for the OP.

Regarding the planned travel, I don't understand the routing. You seem to be proposing to take NZ to Australia from LAX with a connection in New Zealand, but with DL flights from ORRD to LAX. AFAIK DL doesn't fly nonstop from ORD to LAX, so you would be connecting somewhere. This adds risk of missing connections, lost bags, etc. Why not take a nonstop on UA to stay within the same alliance for this trip? Note also that whether you use DL or UA, you need to go land side and change terminals at LAX so be sure to allow lots of time. Alternatively, you can fly Virgin Australia (a DL partner but not an alliance member) nonstop from LAX to BNE, although not every day of the week. VA flies between Australia and New Zealand too. This would be the sensible airline to match with DL, although you could also take DL nonstop LAX-SYD and tack on VA segments to where you want to go down there.