The best areas in Brisbane for expats moving with children are probably Hamilton, Bulimba and New Farm. All of these suburbs are fairly close to Brisbane city centre and Brisbane has a good public transport network so you shouldn't have any problem getting around.
Take a look our page on Accomodation in Brisbane for a better idea of whats available and how to go about buying or renting property in Australia.The Expat Arrivals Education and Schools page will also give you a great overview of the schooling system in Australia.
Best of Luck with the exciting move to Oz !
As a general rule of thumb, the areas around the best schools tend to be family friendly. As you have so much to think about I suggest starting your search by thinking about schools. First decide what sort of schooling you want for your kids - will you be happy with Australian-curriculum schools (the Australian system is well regarded worldwide, and they will be cheaper than private schools) or do you want your kids to get a more expensive, private British education (ensuring an easy transition back to Britain if you return before their schooling's up)?
Once you've made that decision you can narrow down your options based on this list of top schools in Brisbane (http://bettereducation.com.au/school/Primary/qld/qld_top_primary_schools.aspx), where schools are ranked according to their exam results, and there is an indication of whether they are state or private schools. You can look at each school's web site and decide which one would suit you, and then do some internet snooping about the area it's in.
When moving it's always best to start with looking for a school as there can be waiting lists for the really good ones in nice areas.
Hope that's helpful :)