There are two teahouses in Perth. One is called The Tea Merchant, which has an outdoor seating area, right in the middle of the city. It would be a nice place except its notorious for drug dealers. The other is in an inner city suburb called Subiaco, and is far more gentile.
Oh, and there is another called the Indiana Tearooms, which has a fantastic view, perched over Cottesloe Beach, of the Indian Ocean. (Here is an updating webcam view, if you're interested - bear in mind its summer there now:
http://www.coastaldata.transport.wa.gov.au/coastcam/live-cottesloe.html ...and its where, I suspect, Lucifer sat in the pages of Sandman admiring the sunsets). Indiana does a great breakfast, but has a limited range of teas, despite its name."