In my younger days, I used to kayak around Matilda Bay near UWA. I chose this location because there was plenty of parking for my car and the views from the water back to the city were so beautiful. Being tranquil on the water and just watching the cars go by made me feel very peacefull.
There are so many beautiful spots! On the Swan River near Bayswater is always popular, but I've also thought that further up the river past Midland would also be nice. The Canning River near the Canning River Regional Park is also very beautiful and tranquil even though it's completely surrounded by suburbia. Check out this link to find a brochure for the Canning River Canoe Trail and spend a fascinating day exploring this historic and scenic area: http://www.sercul.org.au/docs/CRRP_Canoe_Trail.pdf
If you're down in the Mandurah area there are also heaps of great spots for kayaking, both on the Peel Estuary and on the various rivers that drain into it. Drop into the Mandurah Visitors Centre to get some brochures on the various canoe trails in the area.