I don't know about 'expert," but I've been doing kickboxing for about 10 years.
There definitely is a learning curve involved (even looking at Cathe's kickboxing workouts, her form is much improved from Cardio Kicks to now!).
At first, for me, I had to really think about pivoting my foot and having the power from the punches come from the lower body, through the core, into the back and shoulders. Now, it's pretty much automatic.
I think the only 'trick' involved is prcatice, practice, practice! And slowing down the moves at first until your body gets used to the particular moves and develops some muscle memory (the first workouts I used seemed so fast to me! Then a couple of years later, I went back to use them, just for a lark, and found that I could move faster than the instructor!).
Be patient, practice, and you'll get it.
The slower leg drills at the end of Kick Max should help.
And remember with side kicks, they really aren't full out to the side, but a bit behind the body (which is turned away, so they go to the side of the room). Aim with your butt, not with the side of your legs. The power comes from the glutes (maximus and medius). Keep your heel somewhat higher than your toes and "push" out to the side. Keep the toes of the other leg facing in the other direction, with that heel facing in the direction of the kick.
