n4cer, if you're relying on public transport I'd strongly suggest Sydney Central Hostel YHA, it's right next to Central Railway Station at the southern end of Sydney's CBD. It's also close enough to walk to both Chinatown and Darling Harbour, if you're a visitor on a budget Chinatown's a great place to eat.
Places to see/things to do:
Walk around the Royal Botanic Gardens from Mrs Macquarie's Chair to the Sydney Opera House, there's a red tourist bus which loops around the Sydney CBD, will cost you bugger all. Take your own lunch and have a picnic in the gardens.
Get a ferry across to Manly and spend the day there. Walk around to Shelly Beach while you're over there.
Have a beer in The Rocks at both the Lord Nelson (they brew their own beer, it's fantastic and the counter food's very good) and the Hero of Waterloo. If you go to the Hero on a weekend night very late you'll be entertained by a geriatric group singing along on the piano. Priceless. Make your expedition to The Rocks on a Sunday and visit the street markets.
Get a train up to Kings Cross and check it out. Wander along Victoria Street, Darlinghurst.
Visit the Powerhouse Museum, the Australian Museum and the Art Gallery of New South Wales, if you're into culture, that is.
Lash out and do a BridgeClimb, but check the weather the day before to ensure it's going to be a fine day.
Get a train to Milsons Point and walk back over the bridge.
Get a train to Parramatta and catch the RiverCat back.
Check out Darling Street, Balmain.
Check out Glebe Point Road, you could walk there from central and catch the tram back. Visit the Sydney Fish Markets on the way back.
There are 100s of excellent food outlets in Surry Hills if you want to splurge one night. The Winery, Bentley Bar or MilleVini if you're into wine, Taste or Billy Kwong if you like Asian, pizza e birra or il Baretto if it's Italian or Tabou or Bistrode if you feel like French. Good gastro bubs are The Dolphin and The Clock.
Visit the beautiful Blue Mountains.
