Push-ups are widely used to develop upper body strength, power, and muscular endurance, and are adaptable to different fitness levels. The push up requires a general stiffening of the knee and hip joints, pelvis, and spine to keep the body in a straight line from head to feet, while the shoulders and elbows flex and extend to raise and lower the body, and the scapulae retract and protract to facilitate glenohumeral range of motion.
Set up:
Place your hands on the floor slightly further than shoulder width apart. Externally rotate your hands to engage the shoulder and scapulae.
Execution:
Perform a push up by lowering your body to the floor while keeping your elbows directly above your wrists. Stop the motion when your chest is about elbows height. Reverse the motion by pushing your body up and lock at the elbows.
Coaching tips:
Squeeze your quads and glutes. Keep your core braced throughout the movement (punch in the stomach).
Think of the push up as a moving plank. Protract your shoulder blades while keeping your body in a straight line (shoulders, hips, knees and feet).
Keep your head in line with your torso.
Utilize your downward inhaling upward exhaling breath.

My advice: Start working on slow and controlled repetitions. Choose a regression if you want to make it easier. When you earn the movement add speed, or incorporate other push up variations, such as plyometric push ups or push ups with added resistance.