Why not Sydney, that is my home town, and we have some of the nicet beaches in the world, with great nightlife, shopping and food!
Your travel agent would be the best person to speak to regarding which is best. That way you can factor in availability, price and routing/stopovers.
You can fly into one city, and out of another, it's called an open-jaw. The following airlines all fly to Sydney, Brisbane and Perth. SQ, MH, TG, EK, CX & QF. I'm not 100% sure, so don't hold me to it though. But for example with SQ you could fly LHR-SIN-BNE // PER-SIN-LHR. Then all you would need to do is buy an internal sector BNE-PER. Fares start at around AU$200, which is really cheap for a 4 hour flight!
Enjoy the land of Oz!