I evaluated a company relo offer to Brisbane several years ago. In the end, it didn't make sense for several reasons, chief among them is the provincial attitude in OZ about Americans working in their business culture. Right behind that, though is the fact that the substantially higher cost of living made it unattractive for us.
There are requirements for emigrating designed to ensure the Aussies aren't taking on welfare cases, so that hurdle has to be cleared. There is decent information on their government websites that will help. May also want to engage a local attorney if you decide to move forward.
I didn't spend any time in Adelaide. People from NSW and QLD consider SA to be a back water, but their descriptions of Adelaide painted a picture of a city large enough to have all the services one would need.
I spent several weeks in Brisbane and thought it was the US equivalent of Kansas City - Nice enough, no drama and decent place to raise kids, but far from the things one thinks about Australia.
Sydney I found to be spectacular. I think of it as the business and entertainment of NYC and LA combined, but substantially smaller than either at ~ 4 million people. Horrifically expensive housing. Check out realestate.com.au to get a sense for what you get for your money.
As for the quality of education, I wasn't impressed once I dug in to it (my kids were 5 & 2 at the time). Their public schools ("state schools") suffer from the same issues as many of ours do and I found that going private was necessary to provide the same level of education as was available to us at the time.
One of my colleagues did accept a relo offer a while later and spent 3 years in the Sydney area. While glad for the experience, they never felt their kids were accepted by the locals and were glad to get back to the US.
Some other costs to be aware of:
1. Taxes make CA's look reasonable
2. Food is more expensive than I found it in FL, which is higher than CA
3. Owning a car similar in size to what we typically drive in the US is very expensive. Good luck replacing an Expedition or Suburban down there- won't happen. Forget about importing one either. Can check out your favorite manufacturer's AU website to get an idea.
A few significant upsides:
1. Great, fun loving people who tend to speak their mind, sometimes quite colorfully
2. Spectacular continent and geography. Expect to spend a lot of time traveling to explore it.
3. Strong sense of history and proud of the impact they have had given the relatively small population (~24 million today, much smaller in the past).
Would love to go back for a visit in the next few years with my kids so they can experience it.