Tman - I'm assuming that you are shooting clays based on the part about the instructor and the vest. If that is correct, what kind of shells are you shooting (ie. how much shot, velocity and/or dram equiv., etc...) and what kind of gun are you shooting (auto, O/U, pump, etc...)?
There are a number of different things to try. Here are some generalizations, and we can discuss specifics based on your answers to the above questions.
1) Use the lightest load possible for the task. If you are target shooting, I would try 1 oz light target loads. I know that Winchester makes some very light loads which really limit recoil. If you are shooting an autoloader (especially a Benelli) the light loads may not have enough oomph to cycle the action reliably.
2) Make sure that your form is good on the gun mount. The shotgun should be pulled firmly into the pocket and your elbow extended roughly parallel with the ground. If there is any room for movement, it will hit harder. On the other hand, resist the tendency to have a death-grip. I believe Dave McCracken suggests a grip as if you were holding a small bird you didn't want to have escape. That accurately describes the type of pressure I exert with my hands.
If you are doing the two things above, and are still getting whacked, then something is amiss. It may be poor gun fit, or some quirck with your form. Either way, you want to get it addressed before you start to get beat up enough to develop bad habits (flinching) or a dislike for the sport.
Also, it would be useful for us to know exactly what type of shooting you are doing. If it is clays, what kind (trap, skeet, sporting)?