Bob,
Plans are well afoot to see this part of the world then.


I trust Melbourne will include Puffing Billy.


That will take pretty much of a day (daytime) to do that. But well worth it for a good perve into the Aussie backyards along the train ride (suburban) to Belgrave then the 2 hours or so for Puffing Billy, then return.
I have checked, and Opera Australia are doing a season at the State Theatre in Melbourne the week you are there. With Orpheus of the Underworld on the 20th May, Carmen on the 21st May and Mdme Butterfly on the 22nd May. Tickets are $45 - $177 AUS. My choice from those would be Carmen. Highly recommend it, just for the experience, then, if you're a "gambling man", followed by a bit of supper at the world's biggest casino, just 5 mins walk along the river from the State Theatre.
Sadly, there's nothing of any note (no pun intended!!) on at the Sydney Opera House when you will be there.
Does your travel itinerary include airport/hotel transfers? If not, public transport to/from Melbourne airport is pretty much non existent.
A cab should cost you around $45 - $50 AUS ($22 - $25 US) from the airport to St Kilda. The arival video on the plane should give you a more up to date figure. It's been a while since I caught a cab to the airport in Melbourne. Add about $7 AUS to the cost given in the video, as St Kilda is on the other side of the city from the airport. Once settled into the cab, and on the freeway, the first thing you need to ask the cabbie is "Who won the footy on the weekend?"......... You'll have a friend for life!!
In Sydney? You can catch the train from Circular Quay (just near your hotel in The Rocks) direct to the airport, but a cab should be only about $22 AUS or so.
In Sydney?
A day trip to Katoomba (on the train) is worth it for the scenery. They have the steepest railway in the world (about 500m cable) there, and also a cable car strung across the valley (I've mentioned this before). You can get full day tours from the city, but these are by bus. I think they run a tourist shuttle from Katoomba station, or its only about $5 in a cab to get to the Scenic Railway.
Or this:
On Sunday 25th May, they run this: The 3801 Steam trip down through the Southern Highlands of NSW and return. An all day trip including meals.
click here
I have one of these running (or did run) around Garahbara.

Zoos??? The Melbourne Zoo is a better zoo, but the Sydney Zoo (Taronga Zoo) is much more scenic and spectacular in it's location and getting to/from it (with a ferry ride across the harbour), so I'd recommend that zoo.
Cairns?? There's the Kuranda Railway, and the cable car ride back down the mountain to do there (or vice versa), and, of course, the weather, and Great Barrier Reef to see.
Brisbane? Wouldn't have a clue. Been there a few times, but nothing of note remembering to see, but I'd recommend seeing a game of AFL (Aussie Footy). Melbourne V Brisbane in Brisbane on 31st May (if you are in Brisbane by then), and, of course, I may be in Brisbane on the 4th (Wednesday evening).
You are probably getting all sorts of suggestions as to what to do whilst here, so you probalby wish you were here for 10 years to do it all!!
Let me know if you and your wife have any interests/hobbies/enthusiasms, other than "TOOTS", and I'll see what's going that may be memorable for you.
