Breathing in the breaststroke is simpler than in the other competitive swimming strokes. Nevertheless, there are a few subtleties to be aware of for best performance.
How you breathe in breast stroke depends on if you keep your head above water at all times or if you submerge it regularily like competitive swimmers do.
Let's first imagine that you keep your head above water at all times. In that case, you can obviously inhale and exhale whenever you like. However, it's best to exhale during effort when you push water backwards with your arms.
You also need to be aware that keeping your head above water at all times puts more stress on your cervicles than if you do submerge your head regularly during the breststroke's cycle. Over time this can lead to neck injuries.
If you swim the breast stroke like competitive swimmers do, you must synchronize breathing with the phases when your head is either above or below water.
You inhale quickly when you start to sweep your arms in and your head and chest rise out of the water.
And you start to exhale when your arms recover forward and your head and chest drop back in the water. You continue to do so for the rest of the stroke until you sweep your arms in again. This means that you will mostly exhale with the face in the water.
You should quickly inhale as soon as your head clears the water, and continuously exhale for the rest of the stroke. This will ensure that no water gets into you nose or mouth.
You should not hold your breath, as this might make you more tense, which we try to avoid.
You can inhale and exhale through both the mouth and the nose. I typically inhale through both the mouse and the nose. I also exhale first through both the mouth and the nose. But as soon as I have spit out any water that might have entered my mouth, I close it and exhale only through the nose for the remaining time.
If you have trouble exhaling in the water, you might be interested in doing the basic exercises described in breathing while swimming. Those exercises should make you comfortable exhaling under water.