1. Seeing "Grand" at the Opera House. If you can see a show at the Opera House do it, as it is an experience not to be missed.
2. Visiting Bondi Beach, less than an hour outside the city and completely different. This Aussie surf beach is gorgeous, and it’s fun to watch the surfers and explore the town.
3. Riding the ferries, public transportation at its finest. They may not be the fastest way to get around, but the views of the city are amazing, plus they are relatively quick and inexpensive.
Quick Tips:
When I visited, if you went on an Opera House Tour, there were amazing promos for show tickets (buy one, get one half off, etc.). It worked out so that the tour and my show ticket were cheaper than just buying a show ticket.
The ferry offices offer combo tickets, which include ferry service to a place and admission (I know this is valid for the aquarium), so you save a little money. But, if you are eligible for a student discount, you will do better buying the ferry ticket and admission tickets separately.
Best Way To Get Around:
The Metro is a good way to get around, as are the busses. Ferries are a lot of fun and fairly inexpensive (as I said before, you get a great view of the city and the harbor from the ferry). Walking is an option between many places (i.e. Circular Quay, The Rocks, and downtown are all within walking distance of each other), and taxis are readily accessible.