When teaching side plank yoga pose to beginners I like to use a step-by-step approach. The first step is to learn how to use the shoulder.
To make learning to use the shoulder easier I use a very easy variation of side plank, with both knees bent and the elbow of the supporting arm bent also.
The action is to use the shoulder to lift the ribcage and then relax the shoulder so that the ribcage sinks back down.
This action uses the serratus anterior muscle.
Note that instead of lifting the pelvis, the goal is to only lift the ribcage. That means that there is less weight for the shoulder to lift making this action easier to do.
The next step is to learn how to use the hip. In this case the knees are still bent. After first activating the shoulder to lift the ribcage, next press the bottom knee down into the floor to lift the pelvis.
You may notice a feeling of "muscle activation" at the side of the bottom most hip and maybe even along the outside of the bottom-most thigh. Slowly lower, then relax the shoulder and repeat.
Note that you don't have to lift your pelvis all the way up. You could focus on gradually increasing knee pressure to the point that your hip is just about to lift off of the floor. You could then add just enough pressure to lift your hip an inch off of the floor. Then touch it back down to the floor. Pause before relaxing.
Here's a view from the back.
The next step is to try it with knees straight. The elbow can still be kept bent.
As before, first use the shoulder to lift the ribcage.
Next, activate the feet in such a way that the the blade of the bottom foot presses into the floor. This can feel like you are "stiffening" the foot. Press the bottom leg downwards so that if it isn't already, then the bottom foot presses into the floor. Then lift the pelvis off of the floor. Be sure to keep the bottom foot strong and stable so that balancing is easier.
As with lifting the hips with knees bent, you can first press the foot down just enough that the side of the hip and/or leg only lightly touches the floor. Then lift the hip an inch off of the floor. Pause, touch the hip down, then relax.
Onc you can balance easily with the hip just an inch off of the floor, and once your shoulder and leg are used to supporting the weight of your body, you can work at gradually lifting your hips higher while keeping the shoulder active.
Rather than trying to do the next version of side plank with both knees and elbow straight, we can do it with the elbow straight and knees bent first.
The feeling is slightly different than with the elbow bent and an additional action that you can do after activating the shoulder is to rotate the arm externally so that the point of the elbow points towards the space between the hand and knees (below)
From here, lift the hip by pressing the knee down as before.
One additional action that I would suggest when doing side plank yoga pose with the knees straight is bending the spine towards the supporting arm. So if the right arm is supporting the body then bend the spine to the right as in the picture above.
This seems to make it easier to use the shoulder of the supporting arm.
In the first picture below I have the supporting arm straight and the knees straight but my spine is straight. In the second picture (red shirt) my spine is bent downwards (notice how the right side of my ribcage and waist is more "open.")
Note that also in the first picture my hand is further to the side (with respect to my shoulder). This too can make a difference in how you use your shoulder.
For balance in this position make sure that you keep the feet active and the entire outer edge of the bottom foot pressing into the floor.
To make it easier to use the shoulder to support your body, try expanding your ribcage and lengthening your neck. In addition, externally rotate the supporting arm.
If you feel yourself going out of balance, then move your hips to compensate. If the heel of your bottom foot gets light your weight is shifting forwards. So compensate by moving your hips rearwards.
If the front of your bottom foot gets light on the floor, your weight is shifting rearwards. So compensate by moving your hips forwards.
For more on learning to stabilize your shoulder blades read scapular awareness.
For more on shoulder awareness, flexibility and control read the ebook shoulder yoga.