Whether you just want a relatively short ride or are really adventurous, in my opinion the Munda Biddi track is one of Perth's best rides. Beginning at Mundaring, it traverses bushland through the Perth Hills until it reaches Jarrahdale and then just keeps going south until it gets to Nannup. Eventually it will go all the way down to Albany! If you're a slow rider or just want to 'smell the roses' as you go, there is also simple semi-open cabin accommodation (absolutely free) situated along the trail at intervals or you can bring your own tent. This way a basic bike ride can become a whole weekend adventure. However, if you can only spare a day, doing a short return ride on the track from Mundaring is great and offers some awesome views.
Other good trails are Forsyth's Mill in Sawyer's Valley and the Heritage Rail Trail in the Perth Hills. For more info on mountain biking trails in Perth and further afield check out the WAMBA (Western Australian Mountain Bike Association) website: http://www.wamba.org.au/trails.php#FM