It does depend on what you looking for. If you want a bit of everything then yeah, check out Sydney (the Opera House is beautiful, go for the private tour they hold every hour, it's worth it, Darling Habour and the Harbour Bridge are a must). If you do go to Australia Zoo, it is quite hot there so make sure you bring lots of water and sunscreen (and it is a 2 hours nearly 3 drive from Brisbane and the Gold Coast). If you like party towns and beaches and the touristy type stuff (like the theme parks: Warner Bros Movie World, Sea World, Dreamworld, Wet'n'Wild and White Water World), then you go to the Gold Coast, my hometown, the wannabe Miami!
If you like the theatrical arts, then Melbourne is the place to be. Also if you like wineries, you can go to the Yarra Valley from Melbourne (it's quite cold there, so bring wintery clothes to Melbourne, its known as the place that has 4 season days), also check out the Crown Casino and the Melbourne Zoo, and Werribee Mansion and the Werribee Open Range Zoo.
I guess Uluru and the Outback is probably a must for most, (haven't been out there personally, though I don't have much interest in it). If you want to see Aborigines and corroborees, then you can even go to Currumbin Sanctuary on my side of the country, it's quite a nice spot, has all the Australian stuff that Australia Zoo has plus nice picnic spots and has been there for over 90 years plus you can feed the Lorikeets, a great experience (I perfer Currumbin Sanctuary over Australia Zoo, in case you can't tell already!).