1. First of all, stop using the auto mode and learn to take advantage of the other modes this camera has to offer. If you're going to stick with auto mode, then you might as well have saved some money and bought a basic point and shoot model. The auto mode produces so-so pictures, but not great ones as a rule. And some people complain how the SX20 often produces blurry pictures - one reason is that in the auto mode, it is also in continuous auto focus mode, which can really throw off the focus if you're trying to take a picture of a continuously moving object - like bird, for instance.
2. Setting the mode dial to P is a good way to play with various settings.
3. As mentioned above, turn the continuous auto-focus off, especially for bird photography. You can do this by pressing the Menu button and then finding the option for continuous AF.
4. Don't use the digital zoom. It's better to take the photo at the maximum (20x) optical zoom and crop out the parts you don't want later, using photo-editing software.
5. Generally, I prefer to shoot in vivid color. Seems to be a general improvement over the default, unless you're going for a specific effect. To do this, while in P mode, press the "Func set" button, then choose "My Colors" and toggle this option until you're in vivid colors mode.
6. Do buy a lens cap keeper and use it to keep your lens cap attached to the camera body. One of the few real annoyances about the SX20 is how there is no way to secure the lens cap easily while taking pictures. The keeper has made my life much easier.
7. Keep an eye on the lens hood if you have attached it to the lens by reversing it. It has a tendency to fall off. I have lost one already. Fortunately, you can get a replacement by calling Canon customer support, and it only costs about $15 including shipping and handling.
8. If using the viewfinder rather than the LCD screen (which I like to do to compose my shots better) you may find that the view is somewhat blurry. That is easily taken care of by adjusting the diopter dial (the little wheel to the left of the viewfinder). See this article on how to adjust the viewfinder on the SX20.
9. When using a tripod, turn off the image stabilization mode (IS mode). The reason is that the IS mode tries to compensate for camera shake, and if you're using a tripod, there is no camera shake. So if IS mode is on and the camera is on a tripod with no camera shake, the SX20 tries to compensate for the slightest movement it can detect (e.g., a shaking leaf) and ends up overcompensating, possibly throwing the whole frame out of focus. IS mode can be turned off using the menu button (again, not in auto mode), and then finding the IS mode option.
10. The LCD screen can get scratched up pretty easily. Don't let the screen face outwards when you have the camera slung around your neck. It's a sure way to get scratches from your shirt buttons on to the camera screen.
11. Do invest in a lens cleaning kit that comes with an air blower and lint free cloth. Apart from using it to clean the lens, which should not generally be necessary, you can also use it to clean the LCD display (which can also get scratched from cleaning with an ordinary cloth - I speak from experience here).
12. Post-processing the pictures using photo-editing software can yield quite satisfactory results. I have used the ZoomBrowser EX software that comes on the CD provided with the Powershot SX20 to crop and sharpen some images (using the Edit > Sharpness menu), as well as some of the editing options in Picasa to tweak color and contrast. Here's an before(left) and after (right) example.
Another before (left)-and-after (right) editing example: