Push-ups train the chest, shoulders, arms, core and legs together. If the hips drop, the body may lack trunk strength; if the elbows and wrists hurt, load may be shifting into the joints rather than the muscles.
Build the movement with incline push-ups, knee push-ups, planks and controlled repetitions. Keep a straight line from shoulders through hips, brace the core and stop before technique collapses. Progress gradually rather than forcing repetitions through pain.