Should I inhale or exhale on the exertion? Is it better to have long or short breaths? Do I exercise until I get out of breath?
Just breathe. If you are in a yoga class, meditating, or you are trying to relax and focus, then you should concentrate on your breathing. Otherwise, pay more attention to what you are doing. The goal is the exercise and your performance while doing it, breathing is what happens while you perform, it should not be the primary focus during exertion (other than being recognition of your current limitations).
Think about your posture, think about your movement, be clear about your path of motion, be fluid in what you do, know the beginning and end point of your repetition/range, achieve the proper work and rest periods, then maybe (just maybe) worry about your breathing.
Simply know that the faster you are out of breath, the less you are ready for whatever it is that you are currently doing. On a good day, difficult exercise might not get you out of breath, and on a bad day, simple exercises might get you out of breath. Be under control and clear about what you are trying to accomplish.
If your goal is to get in better condition, it is best to perform MORE exercise without getting out of breath until you are able to sustain significant durations and intensities breathing normally. Then you are free to challenge yourself as it will also be difficult to get yourself out of breath.
Having a difficult time catching your breath on a frequent basis? You’re working above your abilities, not improving them. It’s ok to get out of breath once in a while, it’s a normal part of infrequent exercise challenge, but when breathing or catching your breath is often challenging, improvement in conditioning is going to come slower than you think.