I have a digital camera Canon PowerShot. I notice when I try to take any pictures other than a non-zoom, outdoor picture in good lighting the picture comes out blurry and grainy. I have tried a variety of settings -- but once I move indoors or try to zoom, I dramatically lose picture quality. I have tried with or without flash, etc. But no success. Any ideas?
Low light and zooming affect quality in two different ways.
First low light. Point and shoot cameras struggle in low light because they have small lenses and small sensors. To compensate for this they automatically increase the ISO, which in simple terms is the sensitivity of the sensor to light. The problem is, the higher the ISO, the more grainy (it's called noise) the shot will be.
To get more light in, the camera will also set a low shutter speed, which can introduce motion blur, because the longer the shutter is open, the more movement the sensor will record.
Zooming can decrease quality because it will exaggerate the effects of camera shake. Imagine holding a pencil up to your eye by one end - it's easy to keep the other end steady. Now imagine the same thing with a broom handle - the far end will wobble all over the place. A similar thing is happening when you zoom in your camera - the wobble is exaggerated (not by the increased length of the lens zooming out, but by the more limited field of view).
A tripod can help in both situations, as it limits camera shake dramatically. It won't help if you're photographing moving subjects, though, unless they sit nice and still. As for ISO noise, check your camera's modes to see if you can manually set a low ISO like 200 or 400. Bear in mind that this will have the effect of further decreasing your shutter speed, however.
Fix the ISO to its lowest value and use a tripod. Most small camera will be able to give an exposure of up to a few seconds which is good enough for typical indoor lighting. If it does not do that automatically, switch to one of the Night Scene modes. This will work for static scenes. If your subjects are in motion, you will simply need a better camera.
Small cameras have small sensors which are always at a disadvantage compared to larger ones when it comes to low-light performance. Additionally, they are often fitted with a very dim lens (sometimes called slow) which does not let much light in.
Note that all Canon fixed-lens cameras are called Powershot. There are tons of them but they are not all equal! Some have bright lenses and even slightly larger sensors like the S100 which performs better in low-light. They also now have one large sensor model called the G1 X which would give very good low-light performance. As usual, you get what you pay for, the S100 goes for around $400 and the G1 X about $800. The ones with smaller sensor can be had for around $150 and even less.
Can't add more to EendilTheTall's answer, but make sure your digital zoom is off! Just use optical zoom. Digital can almost never help you. (The only time I'd use it would be when taking a shot with an iPhone that was being uploaded to Facebook right now.)