The NSW Sate Art Gallery in Sydney has regular exhibitions - the Tracey Moffat one which was a travelling show - and earlier there was a retrospective of Jeff Carter's work which was very well done. Also saw an exhibition of significant Sydney buildings a while back. Only problem was they ran out of the accompanying books for the Jeff Carter exhibition and the publisher (to whom the Carters sold the rights some time ago) seems totally disinterested in another print run, nor will they allow the family to reclaim the rights and do a private printing. There are a lot of us with our names on a waiting list!
Then we have the Australian Photographic Centre in Paddington which puts on regular exhibitions of current photographic work, and courses courses, and the National Gallery in Canberra which seems to mount a couple of exhibitions each year.
So maybe it is a Sth Aus thing?