Focusing on your breathing as a correction won't get you very far, as you have already learned. Although utilizing an efficient breathing pattern is necessary for getting a tank to last as long as possible, it is not the primary problem for most new divers.
Simply put, you have to breathe as much as you move. Moving generates carbon dioxide, and the body desperately wants to get rid of that, so the more you make, the more you have to breathe. Look at divers whose tanks last a long time -- my guess is that you will see that they look fairly quiet and smooth in the water.
This starts with good buoyancy control. Neutral buoyancy is a state of needing to do nothing. If you are neutral, you are neither rising nor falling, and in theory, you could hang where you are indefinitely without moving at all. When you achieve that state, you only need to move if you decide to go somewhere, and only as much as you need to get there. Achieving neutral buoyancy is much easier if you are properly weighted, and that's why a weight check is taught in your open water class. If you have not actually verified that the weight you are carrying is correct (as opposed to simply being what someone told you to take) you should do that on your next diving opportunity. You can be neutral and overweighted, but that condition is what I'd call tenuous, in that VERY small deviations from your depth will quickly need adjustment of the air in your BC, because there was too much there to begin with.
Many new divers have no idea if they are neutral, because they never stop to check. If you are swimming, you can hide significant deviations from neutral, because your fins can push you up or down constantly as you kick. So the key to figuring out if you are neutral is the same as the key to breathing less -- STOP MOVING. If every time you quit kicking, you sink, then what do you know? You know you are negative, and making up for it with your feet.
One of the most common patterns in new divers is to assume a partially upright position in the water. It's natural -- we live standing on our feet, and too much diving is taught sitting on our knees, so the new diver's instinct is to try to achieve the posture that feels familiar. There's a problem with this, if you think about it -- if your feet are below your body and you kick, where are you pushing yourself? UP! So how do you avoid swimming back to the surface? You keep yourself negative enough that the tendency to sink balances the push upward. What this means, of course, is that you are using a tremendous amount of energy to go nowhere, and all that energy requires more air. When you can assume a horizontal position in the water, then kicking pushes you FORWARD, which is where you want to go. Then you can truly float, and not only do you use less energy in kicking, but because you are now very stable, you are able to relax and your breathing will naturally slow down. Some of this horizontal position can be achieved with body posture. Try to lie flat in the water, with your head back, and lift your knees as though you were trying to do a yoga posture. If arranging your body this way won't allow you to stay horizontal, then you may need to move some of your weight around, or move your tank. If your feet are dropping despite good posture, it may be that your tank is too low on your body, or that you have too much weight on your belt or in your integrated weight pockets, depending on the setup you are using. Some BCs have trim pockets that allow you to move weight up onto your shoulders, or sometimes you can put weights on the cambands, or tie an ankle weight around the tank neck. Of course, you don't want to go doing these things until you have tried posture first, and determined that you cannot stay horizontal.
Once you have learned just to float and relax, the next thing is to slow down. Many new divers swim like crazy. I think part of it is for the same reason one rides a bicycle faster rather than slower; the faster you kick, the more stable you feel. But the truth is that many underwater creatures use camouflage as part of their survival strategy, and the faster you move, the less you see (and the shorter the time you have to see it in!). Reef diving is basically a lazy man's sport, with a lot of floating mixed with a little swimming. A dive is not the water part of a triathlon!
Once you have corrected the things I've listed, you will find that the rapid, shallow, inefficient breathing pattern that you are using as a new diver will have changed into a more relaxed, slower and more efficient one, without you ever having to think specifically about your breathing.
