Wow, you're gonna love it, Perth is wonderful! I've lived here 16 years and never want to leave!!
http://www.perthcitytourist.com/
Swan Valley: You can spend a whole day here and best of all, it's only about 30mins north-east of the city. There are lots of wineries, a brewery, art galleries, Whiteman Park, cheese factory and chocolate factory, fresh fruit stalls, oggies icecreamery, wagon trails, Caversham wildlife park,
http://www.swanvalleywa.com/ (wagon trails)
http://www.discoverwest.com.au/western_australia/swan_valley.html (swan valley tourism site)
Fremantle: (Sometimes called 'Freo' for short) Historic port town, very 'unique' vibe, kind of bohemian. Lots of old buildings, lots of different museums, cafe strip, buskers, pubs, Freo markets, fremantle prison runs tours, quite a 'touristy' destination, absolutely beautiful!!! About 20min south of the city. Lots of public transport, including a free 'CAT' bus that drives around Freo.
http://www.fremantlewesternaustralia.com/visitfremantle.ASP
AQWA: The Aquarium of W.A (used to be called 'Underwater World') North of the city (maybe 20mins? I dunno, i dont drive.lol). It's part of Hillarys Boat Harbour, which is quite touristy but lovely... cafes, pubs, restaurants, shops, beaches and boats.
http://www.aqwa.com.au/main.asp