It's quite a long drive from Auckland to Wellington, and driving in NZ takes some getting used to -- maybe not so bad for someone like you coming from the UK, but learning about the differences right when you first arrive could be challenging, especially with kids around. So unless you're really into seeing the sights, I would recommend flying. You can book your flights all the way through to Wellington.
I've been to LAX hundreds of times, and it's usually not that bad. It does get unpleasant if you need to change terminals. Most international flights are in the same terminal though, so you shouldn't normally have to change since you would be arriving from the UK. You do sometimes have to walk quite a ways, especially if you have to change airlines. SFO is nicer, but I don't think it's a big deal one way or the other.
One thing to be aware of is that Air NZ doesn't guarantee that more than one piece of luggage per person weighing more than 20 kg will be carried on the domestic connecting flight with you. It's usually not a problem, but if the flight is particularly full, some of your bags might follow later if they're heavy. You might consider packing your important must-have-the-first-night stuff in one bag per person, in the off chance that the other bags get delayed.
You will need to take a little shuttle bus from the international terminal in Auckland to the domestic terminal. It's easy and quick, though.
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