Relaxation is key to proper breathing technique in freestyle. Tensing up and holding your breath while your face is in the water makes breathing more difficult. This forces you to both inhale and exhale when turned to get air -- an inefficient and exhausting way to breathe.
Instead, remember that freestyle is a "long-axis" stroke -- it involves rolling the body along the spine as the hips are rotated from side to side. Breathing should be a part of this body rotation, not an independent twisting movement of your head.
As your body rolls sideways, bringing your face out of the water, turn your head just enough that one goggle lens remains submerged and inhale through your mouth. When you turn back into the water, slowly exhale through the nose and/or mouth, blowing bubbles.
Focus on exhaling air into the water more than inhaling, kind of like reverse breathing. Inhaling will occur naturally as a result of emptying air from your lungs. Practice this reverse breathing out of the water by making a conscious effort to forcibly exhale, while permitting your body to inhale on its own.
You'll quickly learn how deeply you need to breathe to swim efficiently. You'll probably find yourself taking quick breaths during your workouts and taking slower breaths while warming up or cooling down.
Being able to breathe on both sides makes shoulder injuries less likely since you won't repeatedly "torque" the same shoulder each time you take a breath. Breathing on both sides also develops your back and shoulder muscles more equally, and in open-water races, gives you a better view of where you are and where your competition is.
Practice breathing on both sides by taking a breath every three strokes. Breathing on your weaker side will feel odd at first but become more comfortable with practice.
I've been swimming regularly as cross-training for my running. But I know only freestyle. Should I learn other strokes? If so, what strokes do you suggest, and what's the benefit of each?
Swimming is an excellent non-impact sport that provides a full-body workout, but there's more to it than freestyle. Incorporating other strokes in your training will not only improve your conditioning, but it will also better strengthen your muscles.
Switching strokes during workouts will balance the strength of opposing major muscle groups because different strokes use different muscles. Not to mention, swimming the same way session after session can get boring. Mixing it up adds variety to your workout, which may help keep you motivated.
Start by learning the backstroke -- the most similar to freestyle and another "long-axis" body-rolling stroke. Freestyle uses your triceps and latissimus dorsi (or "lats," the large upper-back muscles), with the kick used for balance and forward progression. Backstroke uses the same muscles, but the movement is reversed. The muscles contracting in freestyle are lengthening in the backstroke, and vice versa.
The backstroke is also great as active recovery after a hard freestyle set. An easy swim while floating on your back gives you a "rest" while still moving, and with your face out of the water, breathing is easier.