julesgee you sound like you're in the same position I was in a few months ago! We moved to Brisbane in August (plan to be here around 3 years) and I'm still in the process of getting settled!
Here's what we did, which worked out pretty well:
When still in the UK, we did a lot of research online, looking at the real estate websites that someone recommended earlier. This should give you a bit of idea on what's on offer in your price range.
When we arrived, we took a few days to recover from the trip (otherwise you're dragging a grumpy and tired DC with you, and NOTHING will look good). Then we lined up some visits to houses in a few areas that had been recommended to us - primarily Ascot, Bulimba, Manly and Kenmore.
We didn't necessarily look to rent at first - the rental market moves quite quickly, so if you're not ready to make an offer, then its best not to get your heart set on a particular house - because we too had temporary accommodation in the CBD. But the visits helped us to get an idea of the style we liked, what was available in different areas, etc. Then DH, DD and I revisited the areas (using public transport to get an idea of how a commute would be) and just had a wander, ate lunch in a local cafe, etc to get a feel for the area.
This helped us to narrow it down to Ascot or Bulimba, both of which had the right 'feel' for us and ended up renting a house in Ascot.
Oh, and you do definitely need a car. Public transport is limited. I've also found it difficult to find as many playgroups and activities for my DD (she's 2.5yo) - we're currently on a waiting list for a playgroup
- but there are activities out there, you just have to ask around (and have a car to get to them).
You will enjoy it though. The people are very friendly, the city is a million times more child-friendly than London (or any other British city I've visited) and the weather is beautiful (recent downpours excepted).
I'll shut up now since I'm rattling on, but please let me know if you'd like more about my experience relocating to Brisbane - or someone to call on once you've arrived!