On the jobs front you will find that the majority of work in Adelaide (and Australia as a whole) are got not by what you know but who you know. Top tips for finding work: do a CV drop (more commonly called resumes here) as employers still love this, and also get involved in a local sports club (which will also help you get a local 'personal referee' for your resume).
CVs are pretty straightforward. As a minimum/ baseline, you want to make sure you have:
-list of skills/ attributes
- work history including your main duties (this helps you get past any of the digital resume readers which look for keywords)
- any degrees (including major/minors), licences, trades tickets
You may want to get written refernces before you leave the UK just cos it can be difficult for a prospective employers to contact your UK referees.
Recruitment/ temping agencies are an excellent place to start too. Often you just need a foot in the door.
As for schools, you will have no issue enrolling children in public (state/ govt) schools. Cost? You will need to contact them directly for fees. You may find that some private schools end up being cheaper than public (govt) schools. There's not much difference between public and private at the primary level, but in high school you will tend to find that public schools have a much greater subject offering than private schools in all but the biggest private schools (I've worked as a teacher in both sectors, and a lot of private secondary schools don't hold a candle to the public ones in terms of subjects and even facilities).
If you're looking for a region with a good selection of schools (public and private), the Western suburbs has a very good selection. Be aware that public schools have 'zones'. If you want to be guaranteed your child can go to a particular public school you will need to live in that school's zone. If you don't you will have to apply for a place (not difficult and not often turned down except in schools that are oversubscribed).
When it comes to renting, you will need to pay a bond of up to 6 weeks up front and depending who you rent through, provide evidence of employment. Properties are unfurnished (gumtree and kmart will be your best friends when furnishing on a budget). Apartment/ 2 bdrm living is popular so you might actually find it easier to get a house.
You'll find Adelaide a very easy city to commute around. It's sometimes called the 30min city because most things are within a 30min drive. The "flat" is large. Be prepared to pay more and have slightly fewer school choices of you live in the foothills (Mitcham, Belair, Glen Osmond etc) just because there are fewer families in those areas. As Snifter said, check out different areas. And at different times of day. Nothing quite like a cool afternoon sea breeze on a hot summer day, and nothing worse than an area that traps heat.