Yes, I was a bit taken aback when I saw <<transport for private driver>> and <<budget tips>> in the same sentence.
Do you, perhaps mean that you are the driver and you want to rent a car? If so, that's not a good idea in Sydney - parking is horribly expensive and hard to find.
Sydney is a very walkable city, public transport is good and taxis, while not cheap, are everywhere. There is a free bus that loops around the city that you can use. There are also a variety of travel passes that you can use, depending on where you want to go.
For accommodation, have a look at www.wotif.com and see if you can find something within your price range. Look at Circular Quay, The Rocks, CBD, Darling Harbour and around Chinatown and Central Station. The closer to the harbour, the more expensive the accommodation will be.
I'd agree that you need three days to really get a feel for Sydney and do more than look at the Bridge, Harbour and Opera House. Add another day if you want to go to the Blue Mountains, Hunter Valley or other area outside Sydney.