Thanks for the responses, folks. A lot to chew on.
Based on the feedback here, I get the impression that the inner city isn't family friendly, or it is, but with bad pockets.
What about Maylands, Rossmoyne, Shelley, Riverton, and Wilson? Or Melville, next to Fremantle? They appear to be near the water, not too far from the city, and close to cycling paths in some cases.
The near-water parts of Dalkeith etc are much too rich for my budget, and not too keen on Cottesloe, Scarborough, or City Beach.
Where do the NE suburbs like Yokine and Daniella stack up?
And should I consider looking further north - Sorrento, Ocean Reef, or Iluka? Lots of good property there, although the places appear somewhat dead.
Again, my main consideration is proximity to walks, parks, and basically things to keep a 2 year old entertained with....the "city vibe" is less important compared to that.