I've split Section 3 into two different posts due to the length required to simply scratch the surface of appropriate core training. Next time I'll discuss training the extremities.

Exercises emphasizing trunk/core and stability (and movement):

It’s hard sometimes to dance around the ‘C-word’, because despite its ubiquity, the word itself serves its function well. Say core in regards to exercise, and most people will have a good idea of what you are thinking about (though there are many professionals that couldn’t trumpet the word enough 15 years ago who now throw up a little bit in their mouths when they hear it used today). Since there are multiple opinions of what the core actually encompasses, I won’t try to make a formal definition, but for the purposes of this article, the core loosely refers to the musculoskeletal structures between and including the Pelvis and the thoracic spine and the muscles that act to control their movement.

The main problem isn’t what we call it, but rather our perception of its function. Though the abdominal and back muscles are certainly capable of bending the trunk forward and backward respectively (as well as twisting, and side bending), their primary function is to maintain enough stiffness in the spine so that energy is effectively transferred from the legs to the arms and vice-versa. Imagine playing billiards with a fishing rod. The cue ball won’t go very far, even if you hit the ball straight on. Now play with a cue stick. That is the difference between a stable and an unstable core. To that end, most core exercises should focus preventing motion in the trunk, and difficulty should be increased by placing the body in a challenging position (like a plank) and/or adding extremity motion (like lifting your leg in the plank position).

Not what I had in mind.  Sadly, this image comes from a site 'helping people fix their own injuries' — Not what I had in mind. Sadly, this image comes from a site 'helping people fix their own injuries'

A few things to consider:

To achieve these ends, we will usually start folks with exercises on their back or stomach (dead bugs, plank), then progress to quadruped (pointer dogs) or bridging exercises, then to kneeling, and finally to standing. This progression provides a framework for increasing the weight-bearing, balance, and (number of) joint stabilization requirements of the activity.
It isn’t necessary to spend a lot of time in the less challenging positions, it’s just a good place to start and progress when possible. Most sports aren’t played supine.
There are ways to increase the positional challenge within each of these positions, generally by narrowing the base of support or by adding an unstable surface, but our goal in this section is to increase the amount of force that can be transferred between the extremities, so we’re careful not to limit our ability to address that goal by requiring too much of a balance challenge. I think ideally the challenge will come from introducing some sort of asymmetry or extremity motion, like lifting a leg in the plank as described above.
You can always add resistance, which becomes more practical as you get upright (chops and lifts, cable anti-rotations), just remember the goal is to control trunk movement, not to lift a lot of weight.
Make sure that you are balancing the forces being controlled from front to back and right to left. If you do a plank, gravity is trying to pull you into extension, so the trunk flexors are activated to control it. Make sure that you perform a back bridge so that your trunk extensors get their share of work controlling trunk flexion. If you do a chop, do a lift.
Proper positioning is the best protection against injury. Make sure your clients maintain appropriate postures.
Muscular endurance is better protection against low back pain than strength, but you need lots of strength to pass-block the rushing defensive lineman or push a full file cabinet while maintaining posture. Know what your ends are and keep your workouts specific.
Speaking of ends, remember that unless your sport is competitive yoga, you probably won’t be hold most positions very long. Consider upping the number of repetitions rather than the time a position is held (ie, twelve 10-second isometrics instead of two 60 second isometrics).
Once you add asymmetry by incorporating alternating movements, you might do ok with longer sets because the core musculature, while under continuous stress throughout the set, is constantly changing its firing patterns to adapt to constantly changing demands.
Maximize efficiency. Pushups or inverted rows performed properly are good core exercises, but are rarely used as such, perhaps because we are looking to them for different ends. If you are fit, consider doing them on only one leg, alternating the leg each repetition, and all of a sudden we’re getting more bang for the buck by adding an anti-rotational demand to the exercise.

Once you have demonstrated you know the rules, you can break them.

Once you have developed stability, it might be a good idea for throwing or striking athletes to begin to add core training that includes a rotational movement component. We always want to maintain lumbar stability here, and so a good way to think about incorporating rotation is with a “Top-Down” approach. Initiating the rotational component from the upper extremities will tend to encourage the movement to take place first at the thoracic spine--a good thing, since that is where rotation is meant to occur. A very basic way to introduce this is to go from a pushup position, to a plank on the forearms, and back to a pushup position. The intermediate steps with one forearm on the ground and the other arm straight will create rotation, but because that action originates at the shoulders the segmental rotation of the vertebrae will be biased towards the T-spine. In upright postures, adding rotational movement to exercises like a contralateral push/pull or shops and lifts is also a possibility. In this case, cueing the athlete to ‘rotate from their sternum’ is a good way to encourage the thoracic bias. I am less likely to use an external cue in this case (such as 'twist to face me') because what I am asking them to control is very much internal, and in this case the 'means' are more important than the 'ends'.

As far as incorporating core exercises, many factors come into play. In this particular framework, the core is trained before other strength lifts, but there is more freedom of placement in the workout with core exercise than with any other section or sub-section. It can be trained before other lifts, it can be trained between, it can be trained after. As mentioned in an earlier post, it can be part of the Activation sub-section. It could even be incorporated into conditioning. Just make sure, like everything else, that you always control placement, load and volume to ensure there is not a detrimental effect on either the safety or the effectiveness of more neurologically demanding parts of your program.

Not a selfie....I need to do more plate push-aways.

Section 3: Strength: Part 1: Core Exercises