What time of year will you be here? The summer tends to have more daytime options than winter - although there's still plenty to do in both. I would start the day at a good breakfast cafe - Bondi Beach if you're an early bird, and then walk from Bondi to Coogee on the coastal walk. There's plenty to see around the The Rocks, Darling Harbour, Barangaroo if you like lookouts to the Sydney Harbour, art, architecture & good restaurant options. If you like beaches, Manly Beach, Palm Beach, Balmoral Beach is beautiful as well - take the ferry from Circular Quay for a really nice trip over to Manly. There's always plenty of sports to see on the weekends as well, if you like that, or shopping if you prefer.
Its really dependant on what you're in to - Botanic Gardens Restaurant or a pub session? Rock climbing or high tea? Restaurant/bar scene or fresh food markets? etc...if you want to write a bit about what you're interested in, I can give you more specific suggestions :)

Well basically you go to the beach. That is the reason for Manly's existence. There is a surf beach (South Steye) at one end of the Corso, and a couple of hundred metres west, at the other end of the Corso is a calm harbour beach. Or you can wander around the sea front along an exceptionally pretty walk to Fairy Bower Beach, where there is a calm beach, technically on the ocean but because it faces west is also very calm. Or there is Personally I'd hit the surf.

You can hire a kayak from Manly Wharf and potter around North Harbour. (You used to be able to go para-sailing -- but I think that may have closed down).

Along the North Steyne beach front, you can hire bikes , roller blades, surf gear. You can sign up for surfing lessons.

You can hire snorkelling gear and go snorkelling at Fairy Bower or Fairlight just a short walk away. Or you can learn to scuba-dive, there's a dive shop on Belgrave St.

At the harbour end, there is an aquarium, good but not world-shaking. It is owned by and really has been supersceded by the big one at Darling Harbour.

You can walk up to North Head Scenic Drive for expansive views down the harbour, though it's a bit of a hike.

Or you can do the Manly to Spit Rd, which is a pleasant walk around the harbour foreshore. Then catch a bus back to Manly. (If you're feeling really toey, cross Spit bridge hop along the rocks to Chinamans Beach & Balmoral Beach. Still up for more? Take the Balmoral to Taronga Street then catch a ferry into town and another back to Manly. That should take most of the day.

You should take a ferry into town, because 35 minutes spent on the harbour is 35 minutes well spent.

You can do a ghost tour of the Old Quarantine Station on North Head Scenic Drive, where a hundred years ago they'd isolate you until they were sure you weren't going to bring plague to the colony. Okay, that's actually a night time activity -- don't know if they run daytime tours or not.

You can dodge the The Corso where possible -- well I guess you'll need to eat -- but it's fundamentally a tacky tourist strip.

On the 3rd saturday of each month there's a market at manly Public School which I enjoy. YMMV. There's also a weekly "Arts and Craft Market" on Sydney Road each Saturday -- I'm not familiar with it, I'm sorry, but I'm dubious.