Once a good foundation of basic movement skills has been established, such as being able to use your bodyweight to squat, lunge, push and pull with good technique, posture and balance, then it is good to start challenging yourself with variations to those movements. Sometimes this can involve adding weight in the form of a barbell or weighted vest, sometimes it means lunging up a stair case rather than across a room floor. Before progressing your exercise program or adding movements, be sure to consult with a qualified strength and conditioning coach and your doctor before doing so.
Once my athletes have demonstrated great posture and technique in performing multiple push-ups on the floor, I like to start adding elements to the exercise that will challenge them to maintain that technique. In hockey, very rarely does the body push against a resistance with equal force on both hands or feet; usually one arm or one foot is pushing more than the other. For example, your right arm produces different forces in different directions than your left during a slap shot, or when you skate, one leg supports the bodyweight while the other pushes against the ice to propel the skater forward in a skating stride.
Simply putting a Medicine Ball on the ground and having the athlete perform a push-up with one hand on the ball adds a different element to the exercise, requiring one arm to move differently than the other while coordinating movement. Make sure a ball is used that is solid and stable rather than soft. Usually I will start by having the athlete perform an entire set with one hand on the ball, then another set with the ball under the other hand.
As the athlete progresses, I will have them roll the ball between their hands between every push-up, alternating right and left hand push-ups every repetition. Not only does this challenge the arms to work differently, but maintaining torso stability throughout the push-up challenges the entire torso musculature.
If you have trouble keeping the ball under your hand, you can use something as simple as a low box or dumbbell to elevate your hand higher than the other hand as well.