Disclaimer: I am not an expert on this topic, but like to learn upon and having some guidelines to practice with. As Tamer stated, learning is much better with a teacher. Like lifting.
about half a year ago I read Patrick McKeown's book "the oxygen advantage", which I would highly recommend to get information on the effects of breathing (patterns), the ill effects of (over) breathing, the connection between CO2 and O2 and how CO2 tolerance has big effects on breath. He explains the Buteyko method.
@Steve Freides is a teacher of it.
McKeown recommends a pulse oxymeter, which measures with little delay the oxygen content of arterial blood and heart rate. I got a good one as recommended, and found it quite reliable, as the measurement resonates very well with breath retention time (breath hunger) and the associated sensations.
Usually O2 satiation at rest is at around 97-99%. With breath retention this can be lowered, more quickly with a bit of walking. A measure of progress when I compare McKeown with "the way of the iceman" by Wim Hof and Koen de Jong is both breath retention time.
Today I used the oxymeter out of curiosity:
Deep breathing brings O2 and hr up, retention brings O2 and hr down (CO2 up), breathing after retention brings O2 up and this gets better released from red blood cells because of high CO2 (Bohr effect) content. I have the impression to feel this, as there is a slight tingling sensation.
I think better breathing e.g. breathing less air volume to positively affect body and mind have both methods in common.
I personally really like the WHM breathing method as I read upon by the book. The practice feels good and right to me. I started sitting meditation about seven years ago, also on my own which is essentially just sitting without intention on a cushion for a reasonable amount of time. So I have a bit of a glimpse on breath and heartbeating while doing it.
The last four weeks, I did breathing exercises for 3-4 cycles every morning while sitting after waking up, after some tea. Deep breathing, followed by some retention, followed by just sitting and breathing shallowly. What I can say is, that after such a cycle, breathing for me gets very subtle and my breath likes to have a good pause after exhalation. Currently I have a flex time job at work. When I am sitting on my cushion I am not stressed in any way, that I have to go to work. My mind is very tranquil. In short sitting without much emotion and thoughts is definitely easier for me.
Just my opinion.