When you run, you get out of breath. It's normal, it's natural. Your body needs oxygen. When you exercise, whether running, walking or doing any other physical activity, your muscles need more oxygen. Your body meets this need by supplying oxygen-rich blood to the muscles. Your lungs work harder to absorb oxygen from the air - and you get out of breath.
Without thinking about it too much, most runners breathe in a 2/2 rhythmic ratio. They take two steps as they inhald and two more steps as they exhale. While running very slowly, they often breath in a 3/3 ratio. While running very fast, they might breathe 2/1 or even 1/1. But 2/2 is the most common.
if you count your breaths in and out and discover you are breathing with a different rhythm, don't worry about it. Adjusting your beathing pattern will not make you a better runner.
The same is true whether you breathe through your nose or your mouth. Most runners naturally breathe through both. When asked how runners should breathe, famed New Zealand running coach Arthur Lydiard one replied, "Breathe through your mouth. Breathe through your nose. Suck the air in through your ears, if you can."
Your jaw should be relaxed, your mouth slightly open. The oxygen will come through your nose and mouth to your lungs, to your blood and to your muscles without you needing to give it a lot of thought.
Focus on exhaling instead of inhaling. Your body will breathe in without much help from you. It's a survival mechanism. But just as people sometimes hold their breath while lifting wieghts or doing crunches, many people unconcsciously hold their breath while running. If you focus on breathing out, it helps you maintain your beathing rhythm and your pace. Like breathing exercises used in meditation, it can help to relax you, improving the efficiency and ease of your running. Give it a try, and see if you notice a difference.