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.