The different rail gauges is not true. That has not been the case for 40 years or more.
It all depends how much time you have, and what you wish to see. Flying is quicker for city-to-city transport, but you miss out on a lot by not travelling overland I feel.
I would recommend against coming in May-June, and suggest instead that you come in September-October, which will give you the best balance of low humidity and reasonable temperatures in the tropics, and sun and the beginnings of the warm season in the southern states.
If you were to come for a month, and knowing that you will visit Sydney, Melbourne and Adelaide, I would recommend some combination of the following:
Sydney
Taronga is a full day for a zoo fans. Sydney Aquarium and Sydney Wildlife World will take 3-5 hours, depending on how long you spend at various exhibits. Other smaller attractions in Sydney or surrounding cities are Featherdale Wildlife Park, Oceanworld Manly and the Australian Reptile Park. I've only been to the Reptile Park. To be honest, I doubt there is much there that you haven't seen before. Featherdale might be worthwhile if you want to see lots of native birds. You also want at least a day and a half for non-animal attractions in Sydney. Depending on your budget, a Harbour Bridge climb would be awesome. Then there's Bondi Beach if that's your thing. The National Maritime Museum is something I'm a fan of, too.
Canberra
Do consider going overland between Sydney and Melbourne, with a stopover in Canberra. The National Zoo and Aquarium is a smallish privately-owned zoo that might be good for 3 hours or so. But Canberra is also home to the national Parliament, War Memorial, National Museum, Questacon (a science museum) the Museum of Australian Democracy in the old Parliament House, the National Gallery and various other cultural centres. It's also a good stopover for the Australian Alps, which at that time of year would be excellent for hiking if that's your sort of thing. Taronga Western Plains Zoo at Dubbo is feasible as an overnight trip from either here or Sydney, but you'll need a car.
Melbourne
Again, allow a whole day for Melbourne Zoo. Melbourne Aquarium is probably optional - it's my home aquarium, but I'm well aware that it is not world-standard. Healesville Sanctuary will require booking a coach trip, unless you intend to have a vehicle of your own whilst you're here (which I actually recommend). Healesville is, by my reckoning, the best park for Australian natives. If you're into wine, Healesville is in the middle of the Yarra Valley, one of Australia's top two or three wine regions. A car would also be helpful for Werribee Open Range Zoo, which is much closer to the city but poorly served by public transport. At Werribee, consider the Open Vehicle Adventure tour for a longer, closer look at the hoofstock that are only accessible on the bus ride otherwise. Moonlit Sanctuary probably requires a private vehicle as well, but ask MRJ, its owner, for advice on visiting there.
If you're at all into sport, try to be in Melbourne in the first two weeks of September and try to get tickets to an Australian Football League finals match. If you're in Melbourne between April and August, you'll easily get tickets to a regular season match. If you're here in late September, tickets to the Preliminary or Grand Finals would be difficult to get (impossible in the case of the GF) but still worth spending the afternoon in a pub watching the game.
Ballarat
About 100km northwest of Melbourne - about 90mins drive - Ballarat is home to the Ballarat Wildlife Park, a nice little collection of natives as well as some exotic reptiles. Also consider Sovereign Hill, a gold rush-themed tourist park.
The Grampians
Another excellent national park between Ballarat and Adelaide. Home to the small but rapidly growing Halls Gap Zoo, which would be a good 3 hour visit. But come to this area for scenery, hiking, waterfalls, wildflowers and wildlife.
Adelaide
Adelaide Zoo is third only to Taronga and Melbourne. Monarto Zoo is another open-range zoo about an hour outside the city, and 20mins out of the city is the reportedly excellent Cleland Wildlife Park, another native-focused park.
Kangaroo Island
South of Adelaide, a great place to see wild koalas, kangaroos (if you haven't managed to yet) as well as sea lions, penguins and other sea birds. I think there's also plenty of whales off the coast, although I'm not sure what time of the year that is.
Alice Springs and Uluru
Haven't been here, but obviously Uluru is a mecca for foreign tourists. Just resist the urge to climb it - it's taboo for the local indigenous owners of the site, who don't ban, but don't like travellers doing so. Alice Springs Desert Park, based around native arid region wildlife is supposed to be excellent. Consider taking "The Ghan" train from Adelaide to Alice, and then getting a coach to Uluru, which is "only" 500 or so kilometres away.
Darwin and Kakadu
The number one national park that I've never visited in my home country, but would dearly love to, is Kakadu. An October visit would, I think, be the beginning of the wet season here, so expect an explosion of wildflowers and wildlife. Darwin also has Territory Wildlife Park nearby, a sister park to Alice Springs focused on Australian tropical animals. You can now travel from Alice to Darwin as an extension of The Ghan service.
Cairns, the Daintree, the Reef etc
Probably best as a fly-in, fly-out sidetrip. Cairns Tropical Zoo and Cairns Wildlife Safari Reserve are the biggest of a huge number of small, privately-owned wildlife attractions in this area. Don't miss out on the Great Barrier Reef and Daintree Rainforest.
South-east Queensland
You'll be shocked at how expensive Australia Zoo is for what it has on display. Australia Zoo is probably not recommended as a priority for your trip - depends how 'complete' you want your tour of Australian zoos to be. Underwater World on the Sunshine Coast is apparently Australia's second best aquarium. Brisbane has Lone Pine Koala Park, if you haven't yet had your fill of natives, and the Gold Coast has David Fleay Wildlife Park. It would be remiss of me not to mention Darling Downs Zoo, which is an hour west of Brisbane and owned by Zoochat's very own Steve Robinson. Probably my best recommendation for SEQ would be Sea World - dolphins, seals, polar bears, sharks, tropical fish as well, of course, as rides and other non-animal attractions.
Perth
This would have to be a plane trip across the country. It's 5 hours and not cheap, but they do have Perth Zoo (probably the least noted of the urban zoos in Australia). Margaret River, for wine and surfing, is a few hours south.
You can do all of that in a month, though it would be a gruelling schedule. The "don't miss" trio is Taronga, Melbourne and Adelaide. I would strongly recommend Healesville, Sydney Aquarium and probably Sea World as well. Werribee is worth a visit while you're in Melbourne, and Monarto (though I haven't yet been) is worth a visit whilst in Adelaide as well. Western Plains Zoo is highly recommended, but a long way off the tourist track. Go to at least one of the Australian Alps or Grampians, and live my dreams for me by taking the train from Adelaide to Alice Springs and then on to Darwin.
