There is some good advice from other posters (and some crappy advice too but that has been called out already). Spottygymbag covered a lot of important stuff.
Re: certification of documents - is this for rentals? I have never needed to have anything certified for rentals. More and more agencies are using 1form or similar. You upload all of your documents once and reuse the application for as many properties as use that system (but you can customise it slightly for each property). I suggest you get this set up as soon as you start looking for rentals as the market is very competitive. Agencies that don’t use 1form tend to value personal interaction more so if you are an ‘iffy’ candidate due to lack of rental history or finances etc, you can make a good impression on them at the viewing and then be in the running.
It is really easy to get documents certified in Australia - there are loads of Justices of the Peace who will do it for free. You can Google the ones in your local area and either make an appointment or find out when they have a desk at your local library or mall.
I suggest planning carefully for quarantine, especially for the kids. You’ll want to pack lots of crafts and books etc depending on their ages but bear in mind that you will have to keep your suitcases with you in the room so, at the same time, you don’t want the, to take up too much room.
It gets a lot cooler and windier in Sydney than you would imagine. You will still need lightweight coats. The temperature range is much bigger (sometimes 18 degrees of change in a day) so a cool morning can become a hot day. Layers.
Wollongong would not be my idea of a reasonable commute, even if you have friends there! You might want to try something closer to the CBD.
You said you would be renting out your house - that suggests you might move back to U.K. If so, think carefully about what you want to ship over. Personal stuff, ornaments etc, I understand, but think about your furniture and whether it would be worthwhile or not. You might want to consider selling it in the U.K. and buy new here. I’ve moved a lot and I ship antique furniture but have always found it better to buy mattresses, sofas, soft furnishing etc in the new destination. You might want to price up all new furniture from IKEA and then decide whether shipping anything is worthwhile. Look at Harvey Norman and the Good Guys for appliances and electrics. Also keep in mind that you will need adapters because Australia has different plugs that the U.K. even though both countries operate on 220/240. Again, I buy all new appliances when I move (still really miss my US fridge!).
Shipping your personal effects - clean, clean, clean everything! The shipping company will give you advice on this but I have heard stories (urban legends?) of entire shipments being delayed because of a pair of muddy hiking boots. Australia takes biosecurity Very, Very Seriously!
Good luck with the move!
Is there any more specific advice you would like?