We live in Kensington, just south west of Centennial Park off Anzac Parade. I think Barangaroo is around 30-35 mins by bus from ours, and its only $3.50 (2 pounds) each way.
We have a recently refurbished 2 bedroom unit on a side street, and its pretty quiet, and we pay $600 a week (think this will rise to $630/wk when we renew our lease in July). We don't have a garden, but we have a balcony, and will do us until we have a kid and want a bit of outdoor space. Kensington/Randwick is good (although full of Poms!), there's a couple of supermarkets, gyms, restaurants etc, and only 10 mins car ride (20 mins bus) to the beach at Coogee/Clovely).
Our income isn't so different from yours, and we run two cars (I work at North Rocks to the west, which is an hour drive, or 35 mins via tolls at $15 each way, so I have to be sensible about which way I go sometimes), paying for our wedding in 8 weeks, and some saving for a house. Unlike the UK, we dont spend loads on getting out and doing things, as when you want to get out and do something we just go out for walks/cycle/beach etc.
Hope this helps,
Dan