It's a topic which is rarely discussed, but one which can make a big difference. After all, almost no exercise is improved by holding one's breath during its execution.
-Aerobic exertions (ie, not sprints which are anaerobic, but swims, runs, biking, elliptical, etc) call for a good breathing rhythm. A typical aerobic breathing rhythm is 2:2, that is, breathing in for 2 steps and exhaling during 2 steps. Swimmers know all about breathing rhythm, typically one breath every two strokes or maybe every three strokes, and exhaling with face down in the water. I like to swim with breaths every 4 strokes, but it's just endurance swims, not speed.
- Breathing during sprints (as in HIIT): Sprints are of short duration. Many sprinters inhale once every 10-20 yards. Slow deep breaths, slow exhales. Panting is counterproductive. However you breathe in sprints of any sort, you will not keep up with oxygen needs. You can pant and gasp after you finish.
- Breathing during planks: Planks are typically 30 seconds to 2 minutes. Lots of trainers' advice is slow "in thru nose, out thru mouth," while others suggest in and out both thru nose. The latter is better.
- Breathing with resistance work: As the weights get heavier, breathing patterns become more important. Typically, inhaling during the eccentric motion and exhaling during the concentric motion. (Eccentric and Concentric movements). An exception might be an exertion which demands special core stability, like weighted squats. In such cases, a deep breath can be taken before the eccentric movement begins, and the breath held until exhaled on the concentric stand-up.
Also, with the powerlifts, there is no federal law against taking an extra breath after 5 or 6 reps of an eight rep set, at the end of a concentric motion. Why not? It's not meant to be a sprint. The purpose of weights is just to stress the heck out of muscle to the point of some micro-tears, not to be anaerobic cardio even if it tends to be.
Here's more on the topic of breathing and exercise.