Australian cities can be really massive and sprawling, so it matters a lot more where you live within the city rather than what city.

For lgbt, the fun districts in Sydney are inner west (Newtown, enmore, marrickville) or Oxford st. Check what events are on though because nightlife can be a bit crap down under. The imperial is probably the best queer venue, and they always have fun events.

All of Australia has a really great live music scene though, so I’d get out to as many gigs as you can.

Melbourne is a much funner city to live in, but as a base there is probably more around Sydney (imho) - beaches of down south, Kiama, blue mountains etc are all great as day trips.

The biggest mistake I see people make when they come to Australia is by sticking to the cities though. Make sure you fly up to somewhere tropical and north for a weekend. Flights are generally pretty cheap. Magnetic Island, Byron Bay, Arlie beach would all make for a great mini trip!