Some suggestions (1-5 are best, but I wouldn’t be able to decide on order between them, 6 & 7 would not be as good as 1-5):

1. Sydney Opera House tour: it is SPECTACULAR. The normal one during the day (1-1.5h) is great and a no-brainer. It covers the outside and a bit of inside – but mostly in the public areas. There is also a backstage VIP tour, which is a bit pricey – $150 or thereabouts – but TOTALLY worth it. Starts at 7am, finishes at 9-9:15ish with a breakfast in the Green Room, where all the employees and actors eat. I have done the day one 3 times by now (each guide has its own stories), and the VIP one once.
2. Bondi – Coogee walk: great ocean views, an absolute must-do, whatever the weather. On the Bondi end, try coffee, drink or a meal at Icebergs http://icebergs.com.au/food-drink/ or lunch at Vue if it’s a nice day.
3. Surfing at Bondi Beach: the north end is the best beach in the world to start surfing. And it is full of pros in the south. “Let’s Go Surfing” has lessons that are great for beginners.
4. Harbour Bridge: you can climb it (I have never done it – and don’t have plans), but walking over from Ciruclar Quay to the other side gives you SPECTACULAR views, and takes about 45-60 mins if you take it easy. The first part of the walk is on a side-walk on Cahill Expressway (the highway above Circular Quayt), but it is only 10 mins, and gives a cool view of SOH. You get there with an elevator on the East corner of Circular Quay next to Cahil Expressway (looking towards Sydney Opera House, to your right)
5. Sydney Fish Market: BEST FISH I HAVE EVER HAD IN THE WORLD. PERIOD. A friend of mine lived in Tokyo for years, and was super picky about his fish – he would only eat sushi in 3 places in Sydney. This was one of them.
6. Manly: a suburb in North Sydney, you take a ferry from Circular Quay to get there. A large beach, in a quiet, resort-like place. Tons of good bars / restaurants with a view (and I probably appreciate Ben & Jerry’s there more than US residents will, but still – it’s there if you need it). You can walk to Shelly Beach if you east in Manly at the beach, or to Sydney Harbour National Park if you go West from the ferry
7. Zoo in Taroonga: hands down best zoo in the world we have been to. While we probably appreciate it a bit more because we have kids, it is pretty effing awesome (and has great views of the city when you take a ferry from Circular Quay)

My friend Serena has written an article about her favorite spots too – your can read it here.