I suggest a day trip to Rottnest island. The ferry ride itself will take 1.5 hours. Once you get there, there is plenty to do. Nice beaches to swim at (calm waters), go searching for Quokas (the little marsupial animals on the island), there is a nice playground there with shade cloth, walk around the town - it's quite flat. And hope that your little one falls asleep for the boat trip home!
You might want to hire a car. About $70 a day for a sizeable sedan. The sights of Perth (beaches, town, Fremantle, parks, aquarium, animal parks/zoo etc) are all widespread and it is hard to lug a toddler on and off public transport all day.