How can I get accurate color with my digital camera (Canon A560?)?

When using tungsten, set the camera white balance to tungsten. Don't use flourescent lighting because the colour of it is unstable and shifts between green and cyan constantly. your manual will show how to adjust the white balance and you may even be able to adjust it exactly by using a white piece of paper under the lighting allowing the camera to record the light colour and set the white balance accordingly.

Make sure you are using the absolutely highest resolution available on your camera. I am not familiar with the particular camera you have, but if you have RAW available, use that to get the highest resolution. You then need to follow a system such as the ETTR system (Expose To The Right on the histogram). This will look very bad on your LCD monitor, but will capture the most information available. If you are shooting RAW, the white balance setting does not matter - it is only used to render the image on your LCD which is a form of JPEG. If you use ETTR, most often your LCD will look overexposed.