For this itinerary, I recommend renting a car to get around the city. It will cost approximately $50 for the day including gas. The distance between locations is not too far, and you can easily find street parking or parking lots near each destination.
If you have more time, consider visiting the Angeles National Forest or the Azusa Greens Country Club for a round of golf. If you have less time, consider skipping the Azusa City Library or Downtown Azusa to spend more time in the Azusa Canyon. Also, bring comfortable shoes and plenty of water for your outdoor activities.