A few places I know of...
DeCoursey park is about the best Puyallup has to offer, with maybe 10-12 miles of trails, --now in the spring is the time to ride-- in the summer it gets really overgrown with nettles, and is a mud bog in the winter. Swan creek park at the west end of Pioneer way is OK too, maybe 7-8 miles of pretty technical trail cut into a hillside. A bit farther away is Victor Falls, definetely worth the 20 minute drive out of town.