The most important facet of your shotgunning skills to polish is your gun mount.
You can practice the mount at home -- using a gun you know is unloaded -- with or without an AA MiniMaglite in the muzzle for a 12 gauge; 20s take an AAA MiniMaglite. Turn the beam to its tightest setting so the light shows where the gun is pointed.
First, draw the curtains so the neighbors don't call the police. Start with a low gun, with the butt lightly under your arm and the muzzle below the target, which will be a spot on the opposite wall. I have a print of two jumping wood ducks in my office that works for me. Focus on the spot (for me it's the beak of one wood duck or the other). The first move is with the muzzle toward the target -- in the words of one instructor: "like you were going to hipshoot it." As the muzzle moves to the target, bring the stock to your face and trigger the shot as the butt makes contact with your shoulder. Your head should hardly move at all.
Practice pointing at spots and shooting at crossers by moving the gun along the juncture of the wall and ceiling. Again, move the muzzle first, keep it below that seam, and then bring the gun to your face as you swing past the imaginary bird.
The gun mount does not have to be fast. Speed with a shotgun comes from acquiring the target quickly with your eyes and from shooting without double checking. Moving the gun fast only gets you out of sync with the target.
Practice your gun mounts at home* before moving on to clay birds. When you do, it's very important that you don't move the gun until you see the target. Then, keep your eyes on the bird and trust your hands to put the gun in the right place. It's surprising how simple shooting a shotgun really is.
*Five minutes a night and 20 or so gun mounts is enough to make a difference by the time the season starts. If you want to train like an Olympian, do at least 100 gun mounts a night. Hunting season is approaching; start practicing now.