Aside from the Mana Bars I don't think any city in Australia is all that special for gaming - each has stores that sell games, and a diminishing number of arcades, but that's about it AFAIK.
Every major city here has a zoo: Melbourne's is pretty good, I can't comment on Sydney's as I haven't been there. Melbourne also has a great big aquarium, which I haven't been to in years but remember being pretty good. I think they might have penguins now.
The big thing Sydney has over every other city is the harbour, I guess it's kinda special and you can have your picture taken on a ferry going past the Opera House or under the bridge or whatever. Other than that though, Sydney is a lousy place to get around, it's expensive and I've never thought there was a whole lot to recommend it.
Brisbane will definitely be hot (and, more importantly, pretty humid) that time of year. It's got good beaches, and you're not too far from the Gold Coast / Surfers Paradise with its amusement parks if that's the kind of thing that floats your boat.
Melbourne is probably the best choice if you want to eat / drink / see bands / do other cultural stuff, it's also very easy to get around. The best beaches are a few hours drive out of town and the water is COLD, even in February, but they also have none of the jellyfish etc. problems that you encounter further north (FWIW you'd probably be fine in Brisbane on that count too, it's only the tropical north where the water tries to kill you).
Perth is a really nice laid back city, again with fantastic beaches, but it's also a long way out of your way.
You should be able to get a trains and buses up and down the east coast (Brisbane / Sydney / Melbourne) pretty easily, be aware that it won't be quick though. Sydney is probably a 12ish hour bus ride north of Melbourne, and it's probably another 12 again further north to get to Brisbane.
Edit: domestic airfares are so cheap that it'd probably be more cost-effective, not to mention faster, to fly.
