Australia, Sydney

Take the ferry to Manly for amazing views of the Harbour Bridge and Sydney Opera House. The return ticket price is the same as two single tickets.
Sydney doesn't seem to have a good travelcard travel system. You can buy a day card which gives you travel on all suburban trains, busses and ferries for $15 but a single trip from Kings Cross to the city is only $2 so it is seldom worth it but is a good option for the Manly trip of you will be doing other travel during the day.
Go to the Sydney Aquarium and walk through the glass roofed walkways where you can see the seals, fish, sharks and rays above you. Ask your your backpackers discount.
Take the bus to Coogee Beach and walk to Bondi Beach along the seafront walkway.
Definately plan some time to see the Blue Mountains. We did a day trip which was very short but still worth it. We booked it through our accommodation and it was great. The Featherdale Wildlife Park visit was a highlight. In our opinion better than Australia Zoo. You can get up really close to kangaroos, wombats and koalas without feeling like you're in a Disney park.
Food wise there are lots of food courts in the basements of shopping centers in the city that offer good cheap food.
For transport from the airport check with your accommodation if they offer a free pick-up if you're staying a few nights otherwise there is a door to door minibus service from the airport which costs $10. You may have to wait around until they get anough people but it's straight to your accommodation which is easy with your bags.