October or May/June for travel? I am planning to go to Sydney, Brisbane, Cairns… (Or really anywhere

I want to see GBR and the Australia Zoo, but other than that, I’m flexible and open for suggestions). 12 day trip. What time of the year is best? Even if not October or May.
I don’t want it to be too cold but I am from Michigan USA so “cold” is subjective lol. I’d like to avoid large crowds/peak season. I would like to visit at least 1-2 beaches (Bondi, Gold Coast, Cape Hillsborough, etc…). Right now, I’d really like to stick to the East coast but open to suggestions for another trip to the west.
I have also looked at Kuranda, Daintree, Glow Worm caves. I like to be more nature focused in my travels (beach, mountains, waterfalls, wildlife, etc..)