From what I was told, you can't take pictures while doing the bridge walk so that was a minus. You can get pretty high up doing the walk up the Pylon instead (and its cheaper) - in fact I saw some doing the climb from my perch at the top of the pylon.
The Manly ferry is right by Circular Quay so very close to the Park Hyatt. It's the easier access to the beach. Bondi requires a bus - which is usually crowded at times and just not as cool as being on the water.
I'm not a lie back and relax on the beach type so I'd recommend a beach walk. I didn't do the Manly to Spit Bridge walk, but it looks pretty awesome on the
Sydney website. I did do the Bondi to Coogee one in 2015.
It was a great beach walk with many well maintained walkways - lots of water filling stations and clean bathrooms along the way. You can see a couple of my
pics in my trip report.
For the Blue Mountains make sure you wake up early, it's a ~2 hour train ride there and another 2 coming back. Still a totally neat place to see but figure a whole day, at least, there.
I did a Sydney/Melbourne trip in 2013 (couldn't get tickets/flight for the actual Melbourne cup but managed to catch one of the other race days). But I didn't spend much time in the actual city. I did back to back: a penguin tour at Phillip Island and a Great Ocean Road bus tour (
my pics).
Most people recommend doing the Great Ocean Road tour over several days so you can take your time to explore the towns. I didn't really have time, so the bus tour was great for me.
Phillip Island is about a half day tour - late into the night as they only start coming in when its dark. You cant take pictures of them as any flash will scare them away. They tried a no flash policy but there was always people using one so now everything with a camera is banned. Still pretty cool to see them come to shore into their little burrows. Definitely upgrade to the premiere tickets so you're closer than the regular ticket goers.
I also did a tour of the Cricket Grounds (I actually just wanted the Sport museum, which is pretty cool, but they had a deal for both). Seems you can tour the tennis courts and even play on it if you're interested -
http://www.tennisworld.net.au/tours/