Controlling your breathing is an important part of running, since deep breaths get more oxygen in your bloodstream and muscles, giving you more energy and endurance. While I rarely think about my breathing when I run, I do intentionally focus on it when my breath is shallow, when I experience fatigue, or when I feel a stitch.
Our diaphragm is a muscle that we need to train and use for deeper, more controlled breaths from the belly rather than shallower breaths from our chest. Deeper breaths send oxygen into the bottom of the lungs, increasing the amount of oxygen we can consume and our VO2 max. You can test this by seeing your belly rise and fall when you take deep breaths and your belly remain mostly still when you take shallow breaths. You should also inhale and exhale from your mouth to maximize oxygen intake (or both your mouth and nose). Using just your nose will result in shallower breaths.
The technique of rhythmic breathing coordinates your foot strike with inhalation and exhalation in an odd/even pattern, so you land alternately on your right and left foot at the beginning of every exhalation, when your diaphragm relaxes and you have less stability in the core. This is done to balance the impact stress of running on both sides of your body. Always exhaling on the same foot is similar to carrying a backpack on only one shoulder. In short, instead of an even inhale on 2 breaths and exhale on 2 breaths, try inhaling for 3 breaths ("in-2-3") and exhaling for 2 ("out-2"). Try this while resting, then walking, then running.
Some research also suggests that intentionally holding your breath - for example, during short sprints, can simulate altitude training and allow your body to adapt to lower levels of oxygen. As a result, your muscles become more efficient at extracting oxygen from the bloodstream. If you want to experience this challenge, do 2 sets of 8 sprints of about 5 seconds holding your breath, every 30 seconds. It will be mentally challenging, and you should certainly avoid this if you have any heart, lung, or high blood pressure condition.
Finally, a tip when you do have a side stitch while running: when inhaling, tighten your abs on the side where you feel the stitch for a couple of seconds and then exhale and relax your abs. This will feel like holding your breath then letting go. Repeat this 5-10 times to make your stitch disappear and get your breath back under control!