Make sure you are carrying an extra complete set of clothes in your carryon, lightest for layering and a windbreaker/umbrella. Just in case. And get your partner to carry a complete set of clothes in his/her checked bag, and you as well.
We just returned last Tuesday 9-25 from Canada via LAX from Phoenix and return. We had to do exactly what you indicated in your post:
Check your bags to your final destination
In LAX De-plane, go down to Ground Transportation; go out doors, cross the first street to middle island. At this point you just have your carry-ons.
Above your head facing traffic lane is indication of where you are standing: Note the colors
Rental car Pickup (red) Hotel Pickup (purple) Airport Shuttle Pickups (BLUE)
You want to walk along the loading sidewalk till you get to the Blue area. Look above you to see you are in blue area.
You can ask a security attendant or traffic attendant before you cross the street what shuttle bus you should get for terminal B. or ask a bus driver in the blue area if the bus goes to terminal B or what bus you should take: A, B, or C, or give name of airlines for Aust. or NZ.
Wait for the bus they tell you. Sometimes a bus says All Terminals or All Airlines. Might take longer to get to the terminal you need if you take All Terminals. We went from Terminal 7 to Term 2. I think Term B was just a few stops from where we got on at Term 7. (or Term 2 to 7 on return).
Listen closely to the announcer on the bus; get off either door front or middle.
Cross street, go up outside escalator to Quantas ticket counter (maybe you won't have to do this, I'm not sure you get off on Departure level). Check for your gate on the TV monitors in case of changes. If you are in t he Quantas area, just follow crowd to Security. If you don't have a boarding pass for South Pac., ask agent who is helping people in line where you should stand.
If you already have your boarding pass, go straight to security from the ticketing area. Yes, you will have do all the same things over again. When we dress in the morning of our flight, we make sure everything is off our wrist, neck, waist except our neck passport holder; all jackets are in carryon, etc.
Keep passport and boarding pass out for security checkpoint, then before taking off shoes, make sure you put your passport and holder away, zip shut the purse/satchel, etc. No small liquids, creams, etc. If anyone is born before 1937 you don't have to remove shoes--ask or see sign.
When you return from South Pacific, you will do the same thnig in reverse. Check your bags all the way to Cleveland. If there is no US Border and Customs in the Austrailian/New Zealand airport departure, then yes, you will have to claim your baggage from Baggage area in LAX; you can clear customs in this area, I'm pretty sure, When your bags have cleared customs, you proceed to the Connection counter of your airline; this is usally not too far from the customs area, and re- check your bags for your final destination. We did this many times when returning from Asia (we lived there 13 years). After your bags are re-checked, proceed to the shuttle bus area to go to our final destination airline terminal.
Lugging the large bags and carry on's onto the bus is a pain in the you know what. If you are allowed two bags for check in (at no charge or small fee), then we usually check our wheeled carry on on the return trip, and just carry our "personal" backpack, satchel or small duffel with just enough for the flight (book, keys, cameras, immediate use stuff).
Pauline is right, for a flight to South Pacific, I think from LAX is in the afternoon to Aust. But, if you can, get a hotel in the LAX area the night before your Pacific flight, . It's much easier, especially from the East area. Just that you will have to handle your large luggage to and from the hotel, but, it may be easier in the long run; Make sure you get a hotel with shuttle to airport or cab.
I hope this helps