A) It's great that you're reading the manual and trying, and the fact that you understand histograms will help you even more (the histogram sometimes can be more difficult for people to understand than aperture priority).
First, let's review the basics so we can understand what these settings do. You control the amount of light reaching the image sensor through two basic adjustments: aperture and shutter speed. Aperture controls a diaphragm in the lens that's similar to the iris in your eye. By opening and closing this diaphragm, you restrict the amount of light hitting the image sensor. The bigger the opening, the more light is let in. Aperture is measured in ƒ-stops, which represent a fraction, so the larger the ƒ-stop number, the smaller the opening.
The shutter is a mechanism for controlling how long the light passing through the aperture is allowed to land on the image sensor. The longer the shutter is open, the more light that lands on the sensor. Shutter timing is shutter speed and is measured in fractions of a second (or seconds for long exposures). These two settings are calibrated in cameras to allow you to easily control your exposure. For example, if you want to increase your shutter speed by one step, you'd need to decrease your ƒ-stop by one step.
When you're using aperture or shutter priority, you're using your camera's auto exposure mode; however, you're telling the camera that you want to pick one of the exposure settings and let the camera, in combination with the light meter, choose the setting for the other parameter. For example, you might set the camera to shutter priority (Tv or S) and choose a shutter speed of 1?125 sec. In this case, the camera will always take the picture at 1?125 sec., but the aperture will vary, depending on how the light meter reads the scene. Conversely, if you choose aperture priority (Av or A) and set the camera at ƒ/8, the camera will adjust the shutter speed to the proper setting as guided by the meter.
Why pick one over the other? Because aperture and shutter speed don't control just exposure; they also control the look of the image. If you choose a fast shutter speed like 1?500 sec., you'll stop action in your scene. So if you're shooting sporting events, you might want to set your camera to shutter priority, use a fast shutter speed and freeze all of the motion in your photos. Or you might want the action blurred, so you'd force the camera to stay at 1?60 sec. (or longer). If you choose this mode when shooting water scenes and set the speed to a 1?2 sec. or longer, you'll get that "smooth, blurred water" look.
Choose aperture priority when you want to affect depth of field. Depth of field is the range of distance from foreground to background where the image is in focus. If you set the aperture to a small opening, like ƒ/4, the range of objects that appear to be in focus will be shallow, with the subject and little else in focus. If you close down the iris to ƒ/11, the range of sharpness will increase to include much more than your subject.
Aperture priority, then, helps to control the depth of field. If you're shooting someone outside with a busy background, you might want to have a shallow depth of field so that the background goes out of focus and becomes less distracting. Conversely, you might be taking nature photographs where you want both foreground objects and the mountains in the background to be in focus. In that case, you'd want a large depth of field. Since you probably wouldn't be too concerned about shutter speed, the Av setting would work well.
In short, aperture and shutter priority really allow you to take more control of your camera's auto-exposure capability.