Congrats.
1) If you're just flying point to point from UK to Oz – do you really need insurance?
2) There are some airlines that have "migrant" prices and baggage allowances. Head on over to http://www.britishexpats.com and hit the Australia board. There are a billion threads on it and all more detailed than you will get here.
3) Before you get started, become very familiar with the strict Aussie customs and quarantine regulations. Generally it's not much more expensive to have movers pack everything than to self-pack, and if they pack, insurance will cover damage in transit easier.
Dump as much crap as you can. Get several quotes. Don't forget that you'll arrive in Oz several weeks before your stuff (or will have to send your stuff in advance).
Shipping is cheap; stuff in Australia is expensive. If you have it and it works, it's probably cheaper to ship it than to buy it new.
Company recommendations – probably doesn't really matter, the British mover will just sub it out to whoever's cheapest at the Australian end anyway.
PM me or post here if you have any other questions.