This is my home town (although I have lived here for 10 years)...
Great places to take the kids and some things my DS's loved to do:
Perth Zoo, South Perth, fab place for a day out. Take a picnic as the food is expensive and horrible. They have lockers where you can put your stuff whilst you wander around. You can get a ferry from Barrack Street Jetty (on Perth city side, 10 mins across to South Perth, then a 10 min walk to the zoo)!
Kings Park - botantical garden, has fantastic playground for the kids called "Lotttery West Playground" and 'Stickybeaks Cafe" (great for light lunches, breakfast or just coffee and cake). You can also take a picnic lunch (we often did). 5 mins from city. You can also do a tram tour in/around the city and the trams go up to Kings Park, kids might enjoy this.
Cottesloe Beach (was my local beach), nice little playground near main carpark and cafes across the road! Great to go out for breakfast then take kids to the beach or to the playground.
FREMANTLE
The Fremantle Markets (Fri, Sat and Sun) are good for a wander, food stalls, fantastic fruit and veg (I used to go every week), crafts, Oz stuff. Though a warning go early on the weekend as it gets really busy!
Maritime Museum, good on a wet or really hot day. Not sure on the ages but my DS was 2.5 at the time and was happy to run around, did occassionally look at the boats etc. They also had a theatre which had a puppet show on and that also kept him entertained.
South Terrace has lots of cafes, restaurants etc (just around the corner from the Markets). Go to Gino's for best coffee in Fremantle and they do a fab Italian food. Directly across the road is a fab icecream shop (name evades me at the moment) which our kids loved (can you tell we are foodies). Ciccerellos does great Fish and Chips down at Fisherman's Wharf, kids can look at the boats on the marina. There is also a big park on the way that has a playground as well. Not massive but good for letting off a bit of steam.
Further afield in the foothills is Caversham Wildlife Park - kids can see all sorts of Oz animals (could do a whole day out). Our kids got to touch a wombat and pat a koala. They have a big park area attached with a playground, sandpit and picnic bbq's if you want to do your own. In fact lots of parks have free bbq's so you just turn up with bbq stuff, cook it and enjoy.
Another really nice place to have a bbq is Matilda Bay in Nedlands, kids can paddle in the river/beach area or just run around. Nice shady spot about 10 mins from city. They do have a cafe but food is nothing fantastic. We tended to get coffee's or icecreams. Nice restaurant called "Matlida Bay Restaurant" (we had our wedding reception there) but not sure if it is a great place to take kids and not cheap but one of my favourites! Great views back towards the city and Kings Park.
Well that is about all I can think of for now. Bathtime for kids.
Let me know if I can be of further help 