Remember to keep your body straight, hands not too far apart (about shoulder width), and keeping your arms as close to your body as you can (this is important because if your arms go out to wide it can damage your shoulders). Having a narrow hand placement and keeping your arms close enough to almost touch your sides is what they call military push-ups and are quite a bit more difficult than the way most people do push-ups.
Spread your fingers as much as possible when placing hands on the floor to avoid unnecessary hand/forearm tension and imbalances.
The beginner style of breathing is to inhale while going down and then exhale while going up. INHALE DOWN, EXHALE UP, INHALE DOWN, EXHALE UP.
The Perfect Push Up – Do it right! by Calisthenicmovement