For most subjects I would recommend using aperture priority, as depth of field is the thing you want to control in most cases. However, if your subject is moving, shutter priority is the way to go – you choose the shutter speed you want, and the cameras meter will pick the aperture for a correct exposure.
However, for a moving subject you may need to sacrifice depth of field so you can use a fast shutter in order to stop the motion to achieve a sharp subject. Subject in this sense does not include the background, just the thing that is moving.
For example, lets say that you are correctly exposed with an aperture of f/11 and a shutter of 1/60th of a second. However you are getting too much motion blur in the subject. So you need a faster shutter speed. Increasing the shutter speed to say 1/200th of a second means the aperture will have to decrease to f/5.6 in order to maintain correct exposure. Using a wider aperture will mean you have less depth of field.
So, for a given exposure, an increase in shutter speed will mean a decrease in aperture f-stop number (a wider aperture). You could instead increase the ISO setting if you want to keep the same aperture, but be aware of noise as the ISO increases.