We lived in Australia for 16 years - came back in 2002, for mainly family reasons. We were in south east suburbs of Sydney (Sutherlandshire). If you move to Sydney, try to find somewhere to live which is not too far to the west of the city, otherwise you lose all the good things - beaches, things to do in the city etc. Some of the western suburbs are just acres and acres of new houses on raw blocks. The politics/health service/education/immigration debates are much the same as they are in the UK, and believe me, you can get just as fed up of day after day of hot sun as you can of cold rain!
There are more educational choices - many more Aussie kids go to private schools and as people have previously said, the opportunities for sports are endless. Living in Sydney, you also have the best of the cultural side of Australia too. When I went out there in 1987, there were many complaints that Australia was a cultural backwater but this is absolutley not the case now.
Aboriginal culture is celebrated, but this celebration does not extend to inclusion of aboriginal people. The aboriginal residents of Sydney live in a few small enclaves, not in the best part of the city (to say the least) and there are certain areas which are associated with certain immigrant groups, i.e. Cabramatta is predominantly Vietnamese, Lakemba is Lebanese.
The one thing I miss more than anything else about Sydney is the restaurant scene - excellent, varied, and cheap - we used to eat out 3 days a week.
They do have a national health service, medicare, which is funded by contributions of I think 1 or 1.5% 'tax', which covers the same sort of things as the NHS, and when you think about it, is cheaper than National Insurance. Friends of ours were split 50/50 between those who had private medical insurance to 'top up' and those who didn't bother. Many employers provide health insurance.
Traffic in Sydney can be a nightmare, but there is reasonably good train access from most parts of the metro area into the city. We were lucky and lived right on the edge of the metropolitan area, so could 'go bush' at the weekends. If you like the outdoor life, there is nowhere better - we used to barbecue 12 months of the year, and had many wonderful camping holidays - the campsites, especially those in national parks and reserves, are brilliant. As has been pointed out, when we live in Australia we didn't do the overseas travelling that we do now as it just takes so long to get anywhere, and the 3-4 of the holiday I always suffer from jet lag.