what Crankcase76 said is pretty right. to add from personal experiance:

living in the eastern suburbs makes it pretty easy to get to some of the beaches by PT (Buses)

Around the East and the inner west, the only way to really save much on rent is to rent a house with a bunch of people, and it'll be an old place too (50 to 80 odd years). 2 bedroom apartments below about $400 or $500 a week are starting to get rather old and poorly kept.

My Wife and I lived in a nice fairly large 2 bedroom apartment near chatswood for just over $500 a week. Chatswood is about 20 to 30 minutes out of the city by train.

South of the city there is a lot of new apartment block development, I'm not sure what the prices are like but there has been a masive boom in the area.

Newtown is expensive and kind of bohemian (a bit). also the best coffee shop in sydney.

there are express bus lanes along many of the major roads into the city. Parramatta road, Anzac Parade, and over the bridge from lane cove road. These really cut down travel time into the city.


Traditionally spring and autumn in sydney are beautiful and temperate, winter can get very cold at night, especially in older brick buildings from early last century. and summer can be hot for the same reasons.
lately though it's been cold and wet for the last 4 months. .

Major shopping centres that I am at least a bit familiar with are: Eastgardens, Chatswood (westfield and the chase), Macquarie, Bondi Junction, Parramatta, Miranda and the CBD. there are plenty of others but they are the larger ones that come to mind.

if you're researching public transport times from place to place the cityrail website is very good.
www.131500.com.au

Randwick, Kingsford, Bondi Junction, Chatswood, are all about a 30 minute commute into the city. main stations in the city are Town Hall, Wynyard and Central.

The older suburbs tend to be a bit more tightly packed, especially the inner west, north of the bridge is a little more leafy, as is the western and southern suburbs. Most areas of sydney have parks and sports fields not too far apart.

Where Not to live really depends on what you like in an environment. ie: maroubra is the best swimming beach, Coogee is the sunbathing beach , and Bondi is the shopping beach. Bronte Beach isn't much of a beach but there is good snorkelling/Diving I think. descriptions like that cover all of Sydney.

Parramatta is considered the geographical centre of the city but it's a solid hour from the CBD>
Hope that helps a bit.