This very issue is what caused me to add ":KT" planks into every day of our warm-up for the foreseeable future. I was embarrassed by the inability of people in the gym to do excellent push-ups and decided that the movement will be broken down and retrained until you can do an awesome push-up.
The priorities for the push-up as I see them are, in order: correct and rigid posture, including head position; range of motion; elbow orientation; resistance.
First and foremost, if someone can’t hold his or her body tightly in a straight line from ankle to ear, they’re unable to do proper push-ups. You can kick and squirm and fight this idea all you want, but you can’t escape it. You should be able to visually draw a straight line from the ankle, through the hip and through the shoulder at any point in the push-up, and the head and neck should remain in a neutral position—dipping the chin to the floor isn’t getting you to the bottom of the movement and only makes you look like a saggy hunchback who is likely incontinent.
This rigid alignment should never change throughout the exercise. Don’t push your shoulders up and then later bring your hips up to meet them. This isn’t a push-up and it looks weird. Next is the range of motion. This shouldn’t have to be specified after saying “push-up” but there seems to be some confusion surrounding it. At the top, the elbows should be completely extended and the shoulder blades protracted. At the bottom, the chest should be in light contact with the floor. Again, in both of these positions and everywhere in between, the body should be straight and rigid.
This full range of motion in my opinion should take precedence over the degree of resistance. That is, if you can’t complete a push-up to full depth from the toes, you need to modify it somehow, such as moving to the knees or elevating your hands. Remember when on your knees, you still need to maintain a rigid straight body—the knees simply replace your ankles in this case. The hands can be elevated on a wall, but the wall gets in the way of the face—you’re better off moving the hands to a plyo box or bench so the head can travel without obstruction and make the correct posture possible.
I encourage athletes who can do a few standard push-ups to begin workouts with them and move to an upper body elevated position when you can't achieve full range of motion. This can be difficult on the ego, especially for men, but the benefits are worth any potential embarrassment. No one loads a weight on the bar and benches it halfway down because it’s too heavy (well… excepting board pressing and the like); they start with a weight they can move through the entire range of motion and build up from there. Why the push-up (and pull-up for that matter) are viewed differently, I don’t know.
I prefer the upper arms to be within about 45 degrees from the sides of the body. This allows the shoulder to move through that full range of motion more easily and naturally. Moving the hands wider and bringing the elbows farther from the sides will usually make push-ups easier for people, but it also limits depth for most people and starts tearing up the shoulders pretty quickly.
The push-up is one of those things that when done well doesn’t draw much attention—it’s not a flashy feat of athleticism. However, in my opinion, how one performs a push-up is indicative of that individual’s athletic foundation, and possibly more importantly, how committed one is to excellence in movement and performance. Sloppy push-ups suggest to me a superficial interest in athleticism and a degree of laziness. Put a little attention and effort into the simple things and it will pay returns in the more complicated and interesting ones.
Eat for growth, train for strength, live for the MOMENT.