Lots of good suggestions already... I would suggest staying in Fremantle. It is easy to get around with a little one, and very nice as well. I would suggest going to eat at Little Creatures one afternoon / evening, there is nice food and a good play area for the little one. Also, the Fremantle Leisure Centre has a good pool for kids.
Definitely hire a car, though. Public transport is shocking.
I would suggest Sticky Beaks in Kings Park, Heathcote Reserve in Applecross has an amazing playground with beautiful views, Underwater World is a fun waterpark, the Perth Zoo in South Perth is a good day out. Hyde Park in Mount Lawley is great.
Hillary's boat harbour is probably the most calm ocean for a little one, it is always dead flat and I used to take DS when he was very little a lot. Also, the aquarium is there (AQWA). Cottesloe beach is also gorgeous though.
In terms of shopping, I think Claremont Quarter is fantastic - has everything you need. Karrinyup and the city are also good as well.
Hope you guys have a good time