What is your budget? As
Mwenenzi said, the weather will be better further north, but 4 plane tickets on short notice could be pricey.
Since you are in Melbourne but need to travel to Sydney, you can do a 'triangle' with 2 one-way flights MEL-X and X-SYD, and save time and money (assuming you haven't already booked flights to Sydney). Australian flights are usually priced one-way so unlike the USA you don't pay a premium to book as one-way tickets.
I would suggest the red center. It's quintessential Australia, and that is a good time of year to visit. You can see heaps without hiking. Fly to Alice Springs, check out the
Kangaroo Sanctuary (a must-see but sells out, make sure they have bookings available before you lock in your dates!). Fly to Uluru (or drive/bus, about 6 hours) and see the 'big rock'. It's pretty special. You can fly from Uluru (AYQ airport) back to Sydney.
Here are some photos from my trip to the red center:
https://www.flickr.com/photos/149998...57673508059797
Another option might be Perth, a bit further to fly but the weather is usually nice in late May / early June. Rottnest Island is a main attraction out there, as well as the Margaret River region.
If your budget is tight, I'd stay around Melbourne, especially if you don't mind if its a bit colder. There's lots to see in Victoria, and plenty of threads on that.