As someone who regularly 1. run, 2. runs outside, and 3 runs outside in the very cold weather such as the 27F and 19F wind chill we had here yesterday, my advice is:
1. In warm weather, breathe using the mouth and nose to intake the maximum air amounts you can. But don't gasp for air...your guts will start to ache and you won't process your oxygen properly.
2. In cold weather, chew gum. While chewing gum, breathe in through your mouth and nose, out your mouth. You'll find that most of your intake will be through the mouth. The chewing action elevates your temp and keeps your mouth, sinus and throat warm. Your boogers won't freeze.
3. Control your breathing. Don't over think it, but control it. Maintain a steady pace of breathing. No gasping. If you feel as if you need to gasp for air, slow your running pace for a bit.