If Australia is new then pretty much anywhere will be good.
Bear in mind June / July is the middle of winter so some birds will have moved north out of Australia, or dispersed away from breeding areas and be a bit trickier to find. Still plenty of relatively sedentary species though.
Species you've mentioned will still be around:
(Bush and Beach) Stone-Curlew easier in Queensland though both possible nearer to Sydney
Australian Magpie - everywhere - should get in Sydney easily enough!
Pied butcherbird - more of a dry woodland bird so means heading inland a bit. The related Grey Butcherbird should be findable around Sydney, and Cairns has the Black Butcherbird
Laughing Kookaburra - should be easy enough anywhere, and in Queensland there is the Blue-winged Kookaburra too
Frogmouth - three possible - Tawny possible anywhere and the commonest. Papuan around Cairns, Marbled rare, 2 subspecies in rainforest one ssp near Brisbane the other ssp Cape York.
ebird is used widely in Australia after merging with the local Aremea website some years ago, and is a good bet to research sites.
In Sydney itself the Botanical Gardens is a good spot https://ebird.org/australia/hotspot/L915566
Sydney Olympic Park is another good suburban birding spot https://ebird.org/australia/hotspot/L2555587
And the benefit of a winter trip is if you can coordinate onto a pelagic (dates are Saturday 13 June 2020
Saturday 11 July 2020 at the moment) there is a good bet of half a dozen southern ocean albatross and some other tubenoses http://www.sydneypelagics.info/bookings.shtml#dates
If you make it to Melbourne / Victoria give me a shout!