A family friend asked for a few quick "must see" Sydney spots, what resulted is the following keyboard diarrhea. I guess you could say I fell in love with that place.
1. The coastal walk from Bondi to Coogee (Coogee is where we lived). This is one of the best coastal walks in the world!!! It takes about 1 1/2 hour or longer, depending on how many stops you make. There are about 6 different little beach communities to see, most are equipped with bathrooms and some have cafes. It's on the edge of a cliff and it does have quite a bit of up and down, so bring some water. There is a pretty well marked path, and plenty of local sydneysiders will be doing their exercise on it, so it's easy to follow. Also, keep an eye out for whales. Migration season is coming to an end, but you may see something if you're lucky!
2. Part of this path (just before Coogee) includes a little bay, Gordon's Bay. This is the best little spot for snorkeling in Sydney. Of course, you will need your own gear (there are no operators there) and wetsuits (the water is still cold until Nov or so) but if you are able to do this somehow, it's really great. You may even see the local blue grouper who calls this place his home.
3. Also, not far from this path is the Clovelly Hotel which has a delicious Kangaroo dish, not cheap but the best that I've had in Australia. I'm sure you can find other good spots though.
4. The Royal Botanical Gardens: This is right on the harbour and a lovely park that certainly rivals Central Park (and easily takes it, if you ask me). There are so many freakishly beautiful trees and BATS! heaps of them, keep your ears open for their little squeeks and you'll surely seehundreds of them hanging in the trees!
5. The opera house is a given, but there are some lovely (and fairly high priced) cafes/restaurants just by the big house. It's worth watching sunet here (maybe with a cocktail), overlooking the harbour bridge and the opera house, plus the bats leave the park (which is directly adjacent to the opera house/circular quay) at sunset to explore the rest of the city. It's quite a sight to watch them all fly away!
6. The Rocks - a lovely little part of circular quay, across the harbour from the opera house (15 min walk). There are adorable cafes and shops here, including lots of opal ones! The oldest pub in sydney is here and this area is just generally historic and lovely. There is usually a market on Saturdays.
7. Ride a ferry! The public ferries are quite cheap and run often throughout the day. The ferry ride to Manly (ironic, I know!) is really nice. It takes about 45 minutes to get to Manly, which is a Californiaish beach town, with lots of great ice cream shops. It's worth going there, just for the cheap "cruise ride", checking out the beach, grabbing some ice cream and cruising back. Manly and Bondi Beach are also the best two spots for surfing. Both usually have nice and mild waves with many options for renting boards and even surf lessons. 
8. Walk across the harbour bridge; stunning harbour views! It's looks like a long walk, but it's only about 20-30 minutes. On the other side is an old fashion them park called Luna Park (you can see the big smiling clown from anywhere in the harbour). The park is free, you only pay for what you ride. I think you're kids will love this (and probably you too!!) In the very back of the park, you can pay $10 for all the fun you can have in a giant old fashion fun house..I'm talking magic mirror mazes and sack slides. This place is great!! We've done this several times....walk across the bridge, play for an hour or so in the fun house, then take a short 10 min ferry ride back to circular key. fun!
9. Animals - If you want to see Aus animals there is a wildlife park (right in the city and only takes an hour or two to go through). You can get pretty close to some kangaroos and pet a koala. I also hear the zoo is nice, but haven't been. I wouldn't say this is part of the "real" Sydney, but if you want to sneak in some general Australian animal fun...Sydney can accomodate that!
For me, the "real" Sydney is long meal at a cafe with a good friend, followed by an amazing flat white (latte without the froth) and some quality time with the ocean.
p.s. if you like wine, keep an eye out for Hunter Valley Shiraz. this is the wine region nearest sydney and shiraz is their claim to fame!
