We moved out here in Aug 2009, when DS was 20mo. I brought pretty much everything and still regret what I had to leave behind as we were using it and couldn't envisage living without it for the couple of months between shipping our stuff and leaving.
Bikes and outdoor toys - WASH them very carefully. Any mud and you will be done over by the customs people - they will have them cleaned and charge you for it. This goes for all your shoes and boots, especially wellies as well. I didn't bring any outdoor stuff - couldn't be arsed to clean it all - and had to clean all boots and shoes with a nailbrush to be on the safe side (they WERE inspected).
White goods - buy new here.
Clothes - bring them, especially baby clothes - they cost so much more here and aren't necessarily as good quality. I have been back to the UK several times so far and always brought back clothes with me because they're much cheaper in the UK.
Toys - again, bring what you can because the prices of toys here are more than in the UK. Ditto books, DVDs etc.
I did however bring things like the iron, the kettle, food processor, toaster, vacuum cleaner and stereo - and brought several 4-plug extension leads as well. That way you only have to change one plug (the extension lead one) and it means that anything with a special transformer plug doesn't need a whole new lead buying.
We brought all our furniture except anything with horsehair (amazingly, I had a chair stuffed with that) and only brought the cot mattress (my one extravagance when DS was born) - got rid of all the other mattresses and bought new here, but they were due to be replaced anyway.
Amazon did a free delivery on orders over £25 from the UK before Christmas but that has stopped; M&S had a £10 delivery charge to Australia for a while, don't know if that's still the same. Book Depository do free delivery but don't have such a good stock, I've discovered. Books here can be heinously expensive.
I think linens etc. here are more expensive if you buy the decent quality stuff so bring your own good stuff. I brought all mine, and the pillows.
And yes, for goodness sake don't bring anything "natural" - like untreated bamboo, cane, decorative shells, feathers, natural wood ornaments - chances are they'll be caught by Customs and either impounded and destroyed, or cleaned at vast expense (which may also destroy them).
If you are into natural healthcare, buy Rescue remedy in the UK, it's much cheaper. If you need antihistamine cream, buy it in the UK (you cannot get it at all here). If you like a drystick deodorant for yourself, buy it in the UK - they do one here and it's shit - doesn't keep you dry for more than half a day in the summer!
Don't ship alcohol - they will charge you excessive duty on it. You are allowed to bring in 2.25L per adult (3 bottles of 75cl each) but bring it in your suitcases.
If you're flying out BA (if they're still doing the Australia run), consider getting extra cases - I did that on my last trip back in March this year and it cost me £34 for an extra 23kg, which was fantastic and meant I could bring back my massive haul of G&B milk and almond chocolate which I had bought at 2 for £3
My biggest regret is not bringing my airer. Mostly because it could have doubled as a tent for DS, as my mum's identical airer did for us when we were children - it was needed while we were still in the UK and then was ridiculously expensive to bring airfreight because they charge on volume as well as weight, so we didn't.
Car seats - they're a funny one - they have an extra fixing point in Australia and UK seats just don't cut the mustard so your car seat could be illegal here. Apparently they're still debating whether the Isofix is "safe enough"
and afaik it's not been approved yet.
Bring cold weather clothes - do not be fooled, it gets cold here in winter! We had the open fire going last night, and extra blankets on the beds (Ok, our house has shit insulation but still).
I think that's the full wealth of my emigration experience - if I think of anything else, I'll add it later 