In various sports and even day-to-day life, breathing techniques make a difference in performance. Generally, you want to breathe deeply, steadily, and with your breathing muscles relaxed. Deep, steady, relaxed breathing can be especially difficult during exercise. You may need to adapt your rate of breathing depending on your activity level. Nonetheless if you can breathe deeply and steadily (rhythmically) that will help maintain performance.
It is important not to hold your breath because you put a lot of strain on yourself that way, and you can injure yourself more easily. Ben C. noted the torso tension that can naturally be produced in weight lifting by holding one's breath, but weight lifters will also train to breathe through lifts for the sake of their long-term health and performance. One way to achieve the positives of both regularly breathing + holding your breath is to time your motions with your exhale or inhale. This is common in martial arts and yogas.
You can familiarize yourself with controlled breathing by practicing it with push-ups or other simple movements. Here's an example routine:
- While breathing calmly and with a resting heart rate, enter push-up position
- Begin each push-up movement with an inhale or exhale. Go down with an inhale, go up with an exhale (or vice versa, up to you). Each breath begins slightly before the motion, as if the motion follows the breath / the breath leads the motion.
- Continue exercising this way until your breathing and heartbeat is in an excited state. At that point, relax and try to return to a resting state as soon as possible. Lay down, sit, stand, pace - whatever you need, but especially breathe with your entire lungs (from chest to belly) - to return to a calm, comfortable breath and heart rate.
- Once you're back at a resting state, begin the practice again. As for how many times to go through this cycle, that's up to you, but I recommend you do fewer 'reps' in each session but have more frequent sessions (eg. only go through the cycle 3 times in a session, but have practice sessions many times a week). You can also mix up the exercise with push-ups one day, squats another day, a climbing activity another instance, taichi another instance, etc. As noted, this kind of practice can be integrated into everday-day activities - something as simple as opening a door or picking up & putting down a bag can be an opportunity to practice synchronizing breath and movement.
This routine of controlled excitation then relaxation is used by professional athletes (and was mentioned in a TGO.SE Q&A about scuba) as a way to develop more control over your breath, greater lung capacity, and importantly, more relaxed breathing during times of stress and exercise. As this aids in the performance of martial artists, yoga practitioners, and olympic athletes, I think it will also aid you in climbing.
A specific example of how this would be applied in climbing (disclaimer - I don't climb often, but I do apply this regularly in basic acrobatics and other sports and speak with folks who apply this technique in a variety of activities):
As you begin and work through your climb, maintain a rhythmic, deep breath that uses your entire lungs, from chest to belly. When you reach, swing, push or pull up with hands or feet in a way that takes extra 'umph', make sure you begin the movement with a strong exhale or inhale. You should do this for all movements if possible but especially for moves that take extra energy or courage. You would begin a movement with eg. inhaling - a moment after your inhale begins you would begin the movement - with the inhale and the intended movement completing around the same time.
Throughout your climb, try to keep steady breathing that uses your entire lungs (and accordingly, you'll need steady or smooth movements to match that). You'll find you need to keep your body more relaxed and your motions more rhythmic. The 'relaxed' part is probably the most difficult, because you may tend to tense up muscles in use and put more strain than needed on the main muscle you're using. A key to relaxing is putting minimal tension on muscles - only enough to do what needs to be done - and also working with your entire body to make efficient, balanced movements to 'distribute' muscle tension and keep you most relaxed overall.