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I did the bridge climb in December last year and it was amazing and the 360 degree view at the top was wonderful. Answer
Take a ferry from Circular Quay. Also check out the town of Manly and The Rocks also. If you want to treat yourself a Harbour Brid... Answer
Sarah S is spot on... I would add taking also a ferry from Circular Quay to nearby Darling Harbour and a walk around Barangaro