Composition tutorials too often end up belaboring a concept known as the “Rule of Thirds” and neglect to teach how to make good photos. Here are a few other things I’d suggest considering:
First and foremost, know where your focal point is. You’d be surprised how simple this seems but how often students take a picture of “something pretty” without really knowing what exactly they want the view to focus on. Contrast in brightness/darkness as well as contrast in color will most easily attract the viewer’s eye.
Fill the frame with your subject, while creating a sense of visual balance. Make sure elements in the frame complement each other.
Watch the background and make sure to avoid awkward intersections. I used to belabor this point unnecessarily; now I simply recommend this.
Keep the vanishing point outside the frame to make the space feel larger.
Often-times good photography involves creating depth in an image, both via the lighting on the subject and with the composition of elements within the frame. Consider incorporating foreground elements in your shot.
Also try keeping diagonal lines in the frame to create depth (shoot 45º to a wall rather than against a flat wall).
Does it interact or distract? Keeping the background simple can be a challenge. Something simple like lowering the camera to put the subject against a clean field of open sky can make a big difference, especially with silhouettes at sunset.
Like a good painter, consider how someone’s eyes will move along the frame.
We’re bored by seeing the world from 5–6 feet up. Get creative with the camera.
Bear in mind that when they say shoot below actor to empower, don’t shoot way below unless you want the audience to see it as a creative choice. Subtle height positioning of the camera can give the subconscious effect.
When shooting for motion, you have to consider the interaction of your various shots. Wides to closeups can be ok, but medium shots to closeups will often feel jarring if you change by more than 15mm–20mm
This last point can’t be overemphasized. As in everything, practice will be deliver your largest advantage. The more you experiment and apply, the better you’ll be at finding unique and successful compositions. Try some new things, and don’t forget The Creativator.