Here's my 2 cents worth since I just started shooting trap last year and went through what you are going through right now. It's free advice, and it's probably only worth what you paid for it
1) Buy/rent/borrow some good instructional videos that have the shooter wearing a camera close to his aiming eye. This allows you to have the same view as the shooter so you can see what he sees when he breaks the target. I found this very helpful.
2) Pattern your shotgun with different chokes. I'm not a quick shot, so I use an 'improved modified' which is between 'modified' and 'full'. This gives me a tight enough pattern at the distance I hit my targets. My brother-in-law is a quicker shot and uses a 'modified' choke. A pattern that is too spread out may not put enough pellets on the target to break it if it's not dead on. Sometimes you can tell you hit it but if it doesn't break it doesn't count. You can use a 4' x 4' piece of corrugated cardboard with some kind of bullseye or aiming point the size of a clay at the distance you would hit a clay target. Look for the size of the pattern and pay attention to how the pattern is distributed from where you actually aimed. Is the bulk of the shot below or above or centered on where you aimed? You will have to account for this when you aim at the clays.
3) Buy a box of Fiocchi's tracer shells and shoot at night on a lighted range or on an overcast day so you can easily see where you're shooting. Seeing where you're shooting in relation to where the clay actually is can obviously have huge value. This will help you figure out how to lead the target and will answer the question "where did I miss?". These are $18 for a box of 10 so they're not cheap, but that's the price of an education!
4) Work on your natural point of aim. Your feet should be apart approximately the same width as your shoulders. Aim the gun at a stationary target and lower the gun, then shut your eyes and shoulder the gun with a good cheek weld just like if you were shooting, open your eyes. Where the gun is pointed when you open your eyes is your natural point of aim. Compare this to where you aimed when your eyes were open. The difference represents the amount of body twist you built into your original aim point. The natural point of aim is a good starting point for positioning yourself so you're able to easily swing both left or right to hit a flying clay target. There's no point in starting in a twisted position and having to twist your body even more to hit that crossing clay. Remember to always maintain a good cheek weld. My little league coach used to yell "keep your head down!" when we were batting. It's the same thing here.
5) Experiment with shot size, load, power (dram eq.), and brand of shell. I like 1 1/8 oz of #7 1/2 shot at 3 dram eq. I found that since I allow the clay to fly a little farther than the quick shooters, the larger size of #7 1/2 shot maintains better flight and hitting power at distance. I use 3 dram eq because it was the only power that cycled my semi-auto reliably, and just stuck with it when I got my O/U. Many others use #8 shot and/or lower power. Different brands will have different recoil even for the same shot size, load, and power.
Or you could spend $400 an hour for somebody with a British accent who shoots a $32,000 shotgun in a tweed jacket with a flask of Johnny Walker Blue in the pocket to give you lessons!!
Have fun!!