You can rent furniture (and fridge, washing machine ) until your stuff arrives, at least in Sydney, that's what we did.
You'll have to buy quite a lot of things as you'll need plates, pans, towels, sheets, .... You can find all these things in Target and K.mart. It means you'll have two kettles, toasters, but they are really cheap (less than 10 dollars). I find books extremely shockingly expensive. Yes you can have book depository deliver them for free, but the book price is different. I sometimes buy from Amazon and the delivery is 25$ on average. Mind you, there are items that cannot be delivered here. But library are great here, so I can read to my heart content for free.
Stationary is crap and expensive in my experience in most shops, except Office works.
I haven't find yet shoes I like, so I wonder if that's why so many people walk bare foot? Shampoo and Co are more expensive but you'll find offer every week so it's fine.
We had a suitcase each with mixed clothes. Temperatures can vary a lot and drop 15 - 20 degree in one day. We arrived in september, and it was summer like weather and then mid october, temperature dropped to 6 degree.
One trolley with all the documents as you said, birth certificate, bank, property, .....
If you are taking any electrical with you, buy adaptors in Uk, here they cost 10 times as much.
Be ultra-extra careful with all the requirement to pass quarantine, especially if you are bringing bikes, scooters, gum boots. Leave your vacuum cleaner in UK, it will be a PITA to clean every single particle of dust and it will get massive inspection. Careful with the christmas decorations. The tiniest bit made of natural stuff (pine) can mean the whole box will be destroyed and you'll have to pay 500$ for it.
If you are renting be very careful with the entry inspection report. State every single dent or mark on floor, wall, sink and make 200 pictures of everything.
About what can make it easier for the kids, I can't really answer that, as it's been tough (and still is). More than objects (toys, books,....) it will be people. And this takes times.
Where are you moving?