Benefits of Using Aperture Priority over Programme
It's great. You have progressed from using full Auto to Programme which means you can overide the settings the camera chooses. Now you have taken control, and you can get better exposures.
The next step is to take control of Aperture. This means you can control how much of your subject is in focus. In other words - the Depth of Field
Why is this important? Because how much of your subject is in focus - whether it be a small or large amount. Eg. a shallow Depth of Field would give you a beautiful blurry background like you could use when taking a portrait, whereas a large Depth of Field would allow all the pickets to be sharp if you where photographing down the line of a picket fence or a medium Depth of Field to allow the end ones to start being less sharp, giving the feel of depth. This is the essence of Photography!
Depth of Field changes how your image looks whereas changing shutter speed, other than stilling movement, does not change the essence of the image. Depth of Field can tell a story, give feeling and controls where the eye looks.
So, if you have your camera set on P and don't alter the f-stop, The camera selects the f-stop without knowing what you are trying to create! Hence you don't have control of the look of your image. By selecting Aperture Priority (A or Av) you select the f-stop and then the camera selects the shutterspeed. You can then continue to adjust the expoure, if needed, by using the +/- button as usual which will now only change the shutter speed. Watch out though that your shutter speed doesn't go so low that you can't hand hold the camera without getting camera shake.
You can also use Manual (M) and choose both settings yourself, skipping the stage of using the +/- button.