Sydney's a great place to visit, and you've chosen the perfect time of year. The weather should be superb (if a little hot). Bring summer clothes and a hat and sunscreen.
The public transport is really good (ticket machines are much easier to figure out than Melbourne), so you should have no trouble getting places. As well as trains and buses, a large part of the public transport is the ferry system.
I would recommend you catch a Ferry from circular quay to manly beach and spend a day enjoying the beautiful beach (and have a surf or swim if you're feeling brave).
You can also catch a ferry to the zoo and various other tourist attractions.
The botanical gardens are pretty nice, but no better than any other city's.
The roads are absolute chaos and I'd recommend you don't try to drive anywhere. Lots of one way streets and lots of traffic.
I'm not sure how far out of Sydney you want to travel, but the Blue Mountains are beautiful and make a great day trip (or stay overnight if you can), but it will take some time to get there.
Hope you have a great time! Let me know if you have any other questions
