Before we discuss how you can learn to breathe more efficiently, it's best to note that there are two ways our body has become accustomed to breathing: in the chest/clavicle area and from the diaphragm. It makes sense that chest breathing is shallow and requires more work to supply oxygen to the body. Chest breathing requires a higher heart rate and when used alone, doesn't efficiently engage the lower lobes of the lungs (where the oxygen is taken into the blood). This is easy to do. And, if we don't know any better we may never graduate beyond shallow breathing.
What's the alternative? Enter the diaphragm. The diaphragm is a flat "parachute" muscle at the bottom of the lungs (Douillard, 2001, p. 148). It contracts as we inspire drawing air into the lower portion of the lungs (more efficiently transferring oxygen to the blood). Too, it's also in the lower lungs where carbon dioxide and other gasses are prepared for release. Give it a try: place your hand on your abdomen just below the ribcage. You should feel the diaphragm shrink as you inhale, expand as you exhale. Now alternate between mouth and nose breathing - can you tell which one more effectively engages the diaphragm?
While thinking about changing your habits: consider the nose is created for breathing and the mouth for eating (except in extreme circumstances when the airway is obstructed, of course). If nose breathing is efficient, it engages both chest/clavicle and diaphragm involuntarily getting them to work together in a seamless system. The key is to make this happen both during exercise and at rest. Ever felt dizzy? Then you know what it's like not to be breathing effectively. Mouth breathing allows too much oxygen to enter the system. This abundance of oxygen cannot be exchanged with carbon dioxide fast enough causing its build up in the blood. This can cause dizziness, even fainting.