I stayed in Sydney for 2 weeks when I was backpacking and I stayed in
Wake Up hostel it was a nice hostel with friendly staff and they had a giant room with tv and beanbags to chill on. I honestly wouldnt worry about going by yourself. I went solo and the rooms were I believe 6-8 people.
You'll obviously want to hit up all the toursity spots in Sydney like the bridge, Operah House, Batanical Gardens. This was cool and within walking distance to the Hostel, they had thouands of bats the size of dogs screaming in daylight. Theres a ton of beaches around the city and Bondi beach being the most famous one but it hostely sucked because it was way over crowded.
You can buy the greyhound bus there and get the "Hope on and hope off" option with allows to go to anywhere in 1 direction they offer.
Brisbane city (i believe is only a couple hours bus north) was kinda boring, but they had the Steve Irwin zoo just outside of the city and this is higly recomended. I could have almost stayed 2 days in the zoo as there was so much to do. It's a very hands on type of zoo and educational.
Just south of Brisbane if Byron Bay. I recomend hitting that up and live the beach bum lifestyle while there. Great surf, good night life, beautiful girls and easy to get bud (from Nimbin) If you do decide to go to Byron Bay you MUST go to Nimbin. You'll understand why when you go...
Let me know if you need anymore info. Happy travels!