During my week in Sydney I stayed at Railway Square YHA and Bondi Beachouse YHA. This allowed me to see both the CBD and the coastal areas.
The Opera House and Harbour Bridge
These two landmarks were an easy 40 minute walk form the Railway Square YHA. If you can afford it you may want to a) climb the bridge b) have a meal in Sydney Harbour, I imagine there would be a romantic feel to eating food under the warm Sydney sun while having the Opera House and Harbour bridge in the background. If you have a low budget then you can walk across the bridge for free.
This Beach is a short walk from Bondi Beachouse YHA. The beach is one of the more well known ones in Australia but it is certainly not the best (in terms of sand quality, spaciousness, quietness, cleanliness). If you limited time in Australia then I suggest you do not spend a day sunbathing at Bondi Beach. Instead you may want to do the Bondi Beach to Cogee walk.
Socialising / Drinking
Nightclubs in Australia are expensive, this has prevented me from truly enjoying the nights out here. A day catching up with a friend and drinking cider on the Bondi Beachouse YHA balcony was cheaper and more enjoyable than any of my Australian nightclub experiences.
Here are some pictures I took at sunrise from the balcony.