Okay, so you know how to take photos. Now we need to talk about how to make images!
Learning image composition is about learning how to be a detective. We want to be able to look at an image and understand why it does or does not work. Then we want to be able to decide what to do about it.
Learning composition starts as an analytical process on images that have already been taken. The better one gets at this, the more of it starts happening not to images that have already been taking, but images that are being visualized in one's head, before pressing the shutter buttons.
A note on Visualization
It's sometimes said that photographers take thousands of images without ever pressing a single button. Visualization is the art of photography. The skill of visualization (or "pre-visualziation") often take years for photographers to develop. I believe this is because photographers try to learn their camera settings and perfectly visualize what image will get created when they push the button. That's a hard way to go. The difference between f/8 and f/11 is hard to imagine, and it's very hard to learn if so-and-so shutter speed will affect objects at what-or-what speed.
Instead, photographers should be focusing on the relative. They visualize, desiring, an image with various characteristics, and then finding the settings (and location/angle/etc) that get those characteristics done.
By learning composition before (well, while) learning visualization, the workflow presented is for photographers to visualize individual attributes, like brightness or blurryness of objects changing for their goals. Not just to visualize, but to do something about it.
Basically, the important parts towards crafting quality images.
Then, one works from these goals to get to the right settings to get the photo. "I want this frozen, so I need a fast-enough shutter speed". Learning what shutter speed counts as "fast-enough" is just a matter of practice and, perhaps, trial and error.
What matters is thinking about why an image may or may not work while, and before, taking the photo. One's ability to predict what a camera will do comes second to that. Because being able to predict what an image will look like before you take it is fine and dandy and helpful, but doesn't matter as much as knowing why to change camera settings, or where to go to get the best photo when presented with a scene. Learning composition is how to know this.
The settings don't matter. Only the image matters.
Right then. Let's talk about image composition, and start understanding how images work.