Get the ferry to Manley Island. Have a drink in the harbour hotel.
Afterwards walk through the town to Shelley beach.
The Fortune of War is a busy boozer not too far from the Opera House. Google maps is your friend.
The surf clubs give you free membership if you have your driving license and they're generally a bit cheaper than the tourist places. Having said that, the Aus dollar is struggling so although it's not cheap it's not as dear as it could be.
There's a nice area in the centre called The Rocks and the museum of contemporary art there has a brilliant roof top restaurant.
The fishmarket is amazing and it's heaven if you like to eat fresh seafood.
Definitely get the hop on hop off bus, both routes. It'll take you everywhere.