If you are going to go to just one beach in Sydney (and it's usually still not that warm in October), I would walk between Bondi and Bronte/Coogee. It's a more dramatic landscape than Manly, which is a fairly ordinary beach with a lot of restaurants. Going north from Manly there are some spectacular beaches: Palm Beach, Bungan, Whale Beach.

Botany Bay is not really a tourist destination. On the city side there is La Perouse which used to have a snake man, there is also a very small museum and you can hire kayaks there. Fish and chips are good in the restaurant that hangs over the water. On the other side is Kurnell a major oil refinery, there are some interesting Aboriginal carvings at the Botany Bay National Park.

I would say don't do more than one city.

I think you can see usually see Botany Bay, and the refineries at Kurnell, just after you see the beaches of Sydney's beautiful National Park, out the plane window on the approach to Sydney airport.

I agree with Susan about Manly Beach, but it is iconic, and worth a surf, I think. I also like the short walk from there around Fairy Bower at the northern end of the beach, that's the closest one. If you cross the little beach with no surf at the end, where they do snorkel,(I've seen tour groups there, but just take your goggles and wade in for free) and take the path up the cliff, you will come to crashing surf views down the cliffs. Not as spectacular as the Gap, but easier to get to.

If you can be in Sydney after 18th October, the Sculptures by the Sea Exhibition along the walk from Bondi, would be worth an extra day in Sydney.

http://www.sculpturebythesea.com/Home.aspx