We want to move to Melbourne.
Good instincts!
Most of the state capitals are not very multicultural and are very much Anglo, except for Darwin which has a lot of Chinese and Aboriginals, however the Aboriginals are quite marginalised - racism is alive and well in Australia, but mostly focused on the indigenous.
Melbourne is best for new migrants to Australia, because it's very multicultural. Sydney is also multicultural, but the cultures don't mix very much, they huddle protectively among themselves. In Melbourne, they mix a lot more. Down at the Irish pub we see on the menu "Murphy's risotto." Beef and Guinness - I wouldn't touch it, but it shows the mixing alright.
We host homestay students from Japan, China and Korea. If we went to their countries we'd stand out and be stared at, but here no-one gives them a second look, many are surprised when people stop them and ask them for directions around the city.
Among our friends are people born in Japan, Hungary, Italy and France. At my gym Sikhs work out in turbans next to Australian-born Chinese, and I regularly work in on the bench press with an Indian-born former Victoria Police officer.
At the same time, there exist many ethnic community groups for anyone who's feeling homesick or wants to hear their native tongue.
All this makes it much easier for a new migrant to Australia, there's less culture shock, it's much easier to settle in and feel at home.
The city has many parks - excellent for active people - and also has a number of galleries, lots of public music and so on. There's free entertainment of one kind or another in the city every day. There are as Zarkov said many restaurants, but there are also lots of markets, so for people who like lots of fresh fruit and vegetables, nuts and beans, and do their own cooking, food can be cheap.
For cost of living, the minimum wage is A$14.31 an hour, which after tax is about A$12.40 an hour. So a full-time minimum wage worker would get about A$500 a week.
A typical inner-city rent on a single bedroom flash apartment, or a two bedroom medium-quality unit, would be about A$350 a week. 20-30 minutes on the train further out the rent drops to around A$200-$300 weekly, depending on which suburb you live in.
Utilities are mostly metered by consumption, but if you are a careful user of electricity, gas and water you should be able to keep bills down to about A$30 a week. If you're wasteful and have the airconditioning on all the time obviously it's much higher.
Internet and mobile phone use vary hugely, but you can get 2Mb/s (slower than EU broadband, but fast enough for anyone not a mad file-sharer) for about A$20 weekly.
There's an extensive public transport network which is not well-run, though the operator is changing so we hope for improvements. But for about A$30 a week you can get to and from work if you live within 15km of the inner city.
Cars are considerably cheaper to own and run here than in the EU. Petrol is about A$1.20/lt at the moment.
Fresh fruit and vegetables are commonly A$2-$4/kg, nuts and beans A$5-$10, and meat is around A$10/kg. That's from open-air markets, Chinese-run shops selling in-season produce and so on; add 50-100% for supermarkets and out of season produce. So if you cook at home then you can eat very well for about A$80 weekly for two. Obviously restaurants are more; steak and chips and salad might set you back A$20-$30, for example. A typical fast food meal is A$5-$10.
Of course if you're a bodybuilder eating nine whole chickens and a bucket of protein powder a day things get more expensive
Clothing is cheap because it's all made in China, you can easily get jeans for A$20, shirts for the same. Of course you can buy more expensive ones if you have a job or social life demanding it.
Thus, a person living alone on minimum wage can live quite comfortably on about A$350 a week and save the rest. In practice, the person will probably spend the other A$150 on going out.
A couple's expenses will only be about A$100 weekly more than a single person's since they'll share a home, etc. So A$450 weekly - a couple can live together on a single full-time minimum wage, but they won't go out a lot or have fancy clothes or stuff for their home.
Obviously the minimum wage is for unskilled labourers and the like. Qualified tradespeople and professionals earn considerably more. Friends of mine in IT for a few years get A$65-120,000 annually, about two-thirds of which they keep after tax.
Wages are higher in Sydney but so are costs. When I lived in Sydney for six months I found my spare cash was the same. Wages are lower in other state and territory capitals, and so are costs, too.