Try breathing slowly out through your mouth (or nose, or mouth and nose combined), while your face is pointed down, and be ready to turn your head & suck in air through your mouth as your breathing-side arm starts to come up out of the water. I time it so that it takes the whole time from putting my face down until it's time to get more air, and try to run out right then. If you push it all out early, you'll feel the lungs scream for sure. Otherwise, if you try to breath out and then in, all while your face is out of the water, you'll be gasping for sure, and will never get enough air before your face is in the water again. I also seem to kind of use the back of my tongue to close off my throat a little, to keep from inhaling water. I think I actually have my mouth partly open all the time, using my tongue to close off my throat as needed. As you get the hang of it, your lung capacity will get better, and you'll feel less desparate for air. The more you can relax, the easier it will be.

Okay, to be clearer, here is what I'm saying: I tend to breathe on my right side (my left arm is dominant), so I inhale on the right, starting when my right arm is just coming out of the water, and I exhale slowly through my nose and mouth while that arm is pulling through the water, running out of air just as that arm is ready to come out again. There are other breathing/arm cycles (1 breath to 1.5 arm cycles, which is alternate side breathing, 1 breath to 2 arm cycles, etc), but those aren't important until you get comfortable, and really aren't necessary. It is easier to breathe on the opposite side to your stronger arm, because you have more control while that arm is pulling.

Hope this helps. I do ramble on.