You're welcome. To the best of my knowledge, I second everything Helen has said (but not having kids, I know nothing about schools). The areas around Peppermint Grove, Stirling and Floreat are also supposed to be nice if you can afford them. Bits of Innaloo are okay too, and you're closer to the train at Stirling. If your main criteria is short term, Scarborough should have options (plenty of self-catered holiday apartments there). The only other thing I can think of is to do what my OH did (he came over first), and find a rental through a private landlord as they may be more flexible than an agent with their minimum contracts. He found the house through an ad in the West Australian, but you may have some luck with Gumtree.

For public transport, if you can walk to Scarborough Beach Road or Hale Road, there's loads of buses and some go through the suburbs as well. However, I found that if you don't drive, you're kind of stuck for a few things. For shops, apart from Innaloo, there are a lot of IGAs dotted around the area (City Beach, City Beach/Wembley Downs, Doubleview), so wherever you are there should be one in walking distance or a short drive away.

Personally, I really liked Wembley Downs, despite the annoying proximity to the neighbours. It was quiet and I used to catch the bus to work (lucky the office was near a train station as the buses are only really there to connect with the trains), and as I walked down to the bus stop, strangers used to say hello. There were a lot of retired people there who used to go for walks before 7 in the morning. Lots of parks/ovals/open spaces to make up for the small plots. I only ever met people through my OH or work, so I feel for you moving without knowing anyone. You'll meet people through your daughter (whatever she's in to there will be some kind of club), or work, or chatting to neighbours, or there's the bowls club in Doubleview. In my experience, the Western suburbs are a great place to start off in.