Yeah, what most people do to breathe is just raise and lower the upper part of their chest. That's not how an athlete breathes. First of all, forget about how it looks. Every now and again, you might see a photo of one of your champions doing this, but mostly the photogs pass these ones by and pick the normal-looking ones.
Anyway, doesn't matter if it's through the nose or mouth to begin with. So as you start to breath in, forget about your chest - hold that still. Expand your belly out until it's as far out as it goes. Then move the center of expansion up your chest until it reaches the top. You'll look pregnant. I don't think there's a method for letting it out - just exhale forcefully. Repeat. Next time you're on a climb and it starts picking up, start doing this. Keep doing it as your breathing becomes faster and faster. It's relatively easy to maintain at LT, though you might sound like a steam engine. That mostly goes away as you get better. When you go anaerobic, it's hard to keep doing that and most folks just start panting. But maintain the belly-breathing as long as you can. You'll see that you really have to open your throat to move that much air, but moving air is what you have to do to increase VO2max.