Breathing is vitally important with both resistance training and cardiovascular exercise.
With weight training, breathing incorrectly can significantly increase blood pressure during the exercise. This can be dangerous as blood pressure readings of over four times the normal rate have been recorded!
When performing a resistance training exercise, breath out on the concentric phase and in on the eccentric phase. The concentric is where you are contracting the muscles being targeted, or the harder part to the exercise. So, for example on squats, concentric is as you are pushing up and the eccentric is on the way down. For bench press, the concentric phase is as you are pushing up and the eccentric phase is as you are lowering the bar down to your chest. For lat pulldown, the concentric phase is as you are pulling down and the eccentric phase is when you are returning to the starting position.
With cardio training, breathing correctly is imperative to deliver oxygen to your body. By taking short, superficial breaths, you will not deliver a substantial amount of oxygen to the muscles being worked. As a result, your output will be significantly inhibited. Try to focus on deep and controlled breathing, no matter what the intensity of your cardio session may be.