For running i believe you are supposed to find a rhythm to your breath depending on how you are running. So breathe in X steps out X steps. An example is I breathe in for 3 steps and breathe out for 2 steps when running moderately, but when running fast it is breathe in 2 steps and breathe out for 1 step. I usually breathe in through both mouth and nose at the same time since the nose doesn't supply me fast enough then breathe out through both. Supposedly, a good breathing ratio also helps prevent injury by balancing your body, not sure about this though.
For cycling i need still need to find my rhythm!
When you run, you get out of breath. This is normal. This is natural. Your body runs on oxygen, just as your automobile runs on gasoline. When you start to exercise--whether running, walking or any other physical activity--your muscles need more oxygen. The body meets this need by supplying oxygen-rich blood to the muscles. The lungs work harder to absorb this oxygen out of the air.
You get out of breath.
Without giving it much conscious thought, most runners breathe in a 2/2 rhythmic ratio. They take two steps as they inhale; they take two more steps as they exhale. While running very slowly, they often breathe in a 3/3 ratio. While running very fast, they might breathe 2/1, or 1/1, but 2/2 is much more common.
If you count breaths in and out and discover you are breathing with a different rhythm, don't worry about it. Adjusting your breathing pattern will not make you a better runner.
The same with whether you breathe through your nose or your mouth. Most runners naturally breathe through both. Famed New Zealand coach Arthur Lydiard, when asked how runners should breathe, once 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. Breathing is a very natural activity--and so is getting out of breath when you run.