Australia has a number of different visas (length/purpose of visit/country eligibility), ETA subclass 601, eVisitor 651, and Visitor Visa 600. eVisitor is eligible for certain citizenships (U.S. citizens are not included). Visitor Visa is for if you want to stay longer than 3 months at a time. The last one is called the ETA.
Day 1
-Take the train from the Airport into the City
-Walk around Darling Harbour
-Ferry to Circular Quay (Purchase the Opal public transport pass)
-Take a walk through the Royal Botanic Gardens
-Sydney Opera House Tour
Day 2
-Ferry to the Taronga Zoo
-Trek to Chinatown for some great food at Kiroran Silk Road Uighur Restaurant
Day 3
-Walk along side Esplanade street
-Visit the Cairns night market
-Spend some time at Cairns Central Shopping Centre
Day 4
Day 5
Day 6 and Day 7
Day 8 - Day 9
-Eat at Vivo Bar and Grill
-Pick up breakfast the next morning at Rising Sun
-Fly back to Sydney
-Train to Newtown and dinner at Thai Pothong
Day 10
-Breakfast at Cornersmith in Marrickville
-Farmers Market on Addison Rd and Millawara Rd on Sundays
-Train and Bus to Bondi Beach
-Bus to The Rocks