Apply for your Australian ETA visas at least 8 weeks before travel to avoid any problems.
https://www.eta.immi.gov.au/ . For a typical US citizen an ETA will work and be approved within 24 hours, but there are exceptions so best to apply early in case you find out you need another visa. Since it is good for 12 months, no real issue applying early. It is not a transit visa but the easiest and cheapest way to get into Australia. It will allow multiple entry, so if you did have to enter Australia on the way out (say due to a flight cancellation, etc), you can.
Your outbound should be straightforward, your bags should take care of themselves from DTW all the way to AKL. Just make sure they are tagged all the way to AKL when you check them in Detroit. You won't have to go through immigration until you arrive in NZ. Review how the 'smart gates' work in NZ before you arrive, because you might be a little out of it after that long journey!
On your return trip, you'll obviously enter Australia at SYD using your ETA visa. Then when you arrive in LAX, you will go through immigration and pick up your bags, before going back through security for your final flight to DTW.
I'd advise you not to bring much, if any, food products into Australia or NZ. If you do, you'll need to declare them. If you accidentally have some leftover snacks from the plane, just dispose of them in the quarantine bins before customs so you won't have to declare them. There will be sniffer dogs, and they find everything!