As everyone else has said, It depends where in Brisbane that you live. Like every other city, the closer you are to town the more expensive the rent. Living in the outer-suburbs of Brisbane you will have no problems finding a house to rent between $350 - $800 (expect AirCon, pool, large open living areas) a week. Every city has it's less desirable areas which I won't mention so as not to offend anyone, but I'm sure an Estate Agent will be very honest with you on where to perhaps not settle!
Brisbane Public transport is great, although expensive depending again how far out of town you live and even driving is not as busy as the UK, coming from someone who drove in London for 9 years it's absolute bliss!!I will warn you QLD Aussies, of which I am one, are horrendous drivers so ensure you're on a train or bus route!
We moved back to Oz (Sunshine Coast) after 9 years in London and found that the cost of living here is perhaps on par with that of the UK. You don't have as many choices for Supermarkets & Utility Companies hence the higher costs. However, sometimes it cheaper and easier to shop 'Hunter-Gather-Style' as we call it, at the local Fruit shop, Butchers, Bottle Shop (liquor store as we don't sell alcohol in supermarkets here in QLD) & then just get your necessities from your local supermarket. Aldi has made it here on the Sunshine Coast & I must admit, I'm no longer an Aldi-snob as over here in Oz they're pretty good stores I would say our grocery bill from Sainsbury was probably cheaper than over here due to the competition and ease of 'imported' goods.
Anything grown, built, designed outside of Australia is way more expensive (for example Ikea is 3 times the price of the UK) ..... leading me to this final suggestion........ When deciding what to pack and what to leave behind/throw away... bring everything! This is only our experience, but to replace like for like over here was very costly and to find the same quality of goods from the UK in furniture, tools, kitchen goods, sporting equipment was a costly exercise & we felt it would have been cheaper to bring the lot!!
Maybe not everyone's experience but definitely ours & it makes life much easier when you're settling to not feel like you're constantly haemorrhaging cash!
Best of luck with the move...