Fremantle and its associated sights is nice for a day trip, especially if you can co-incide it with a Fri/Sat/Sun when the markets are open and there is a lot more atmosphere. It has a lot of cafes, restaurants and ice-cream shops, a big park called the Esplanade for the kiddies, some historical things like the Fremantle Prison which is worth a visit or the Roundhouse, some gorgeous old buildings, lots of pubs, a fishermans wharf with yummy fish and chips and a few shops and boutiques. I love Freo.
Kings Park gives a nice view of the city and there are a few playgrounds spread throughout it as well.
You could take a day trip across to Rottnest Island.
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