I.Hartman here is some advice I gave to Bob last year, plus some addition thoughts. I'm going to cover all the basics, step-by-step, assuming you don't really know anything about your gun yet.
The first thing you have to determine is where your gun shoots. When you are satisfied the gun shoots straight, you'll want to determine where you shoot it.
Set up on a bench with a target between 13 and 20 yards in front of you. If you have interchangeable choke tubes, you'll first have to determine whether they all shoot to the same place. Pick a barrel and put a choke tube in it. Sight the shotgun just like it was a rifle and fire at your POA. Do this at least 5 times, or enough so that you can say with authority this choke prints 1" high and 1/2" right of POA. Now do the same for every other choke. Make sure to use the same barrel for all of these tests. At the end of your choke testing you will know which chokes shoot to the same POI and which don't. Mark or discard the ones that don't.
If your new gun is an O/U, put your two tightest good chokes in and repeat the test. That will tell you if both barrels shoot to the same POI. If a combo, all three barrels should shoot to the same POI with the same sight picture and setup.
Once you know your hardware is good, it's time to see where you shoot your gun. I like to work on one variable at a time. Any distance from the muzzle to target between 13 yards and 20 yards will work. I prefer 15 yards most of the time for reasons I'll not get into here. I use 40" wide paper and draw three vertical lines on it about 12" apart. Then I mount the gun while standing just as I do on the line, and fire while tracking a line. I find I'm much better if the gun is moving, so I just trace the line with my bead and fire whenever. I only care about right and left here. If I'm consistently dead on the line, I'm done. If not, I move the comb right or left until I am.
Next I repeat the process using horizontal lines. Here I am trying to set my POI to the proper height. I know what my POI needs to be because I have already established it through experience. If you have not, then set you gun up so it shoots about 2" high at 13 or 15 yards, then go and do as hmb suggests. Lock the trap to throw straights and shoot a bunch. If you are smoking birds and you like your sight picture, move to Station 1 or 5 and repeat. If you are not happy, adjust your comb to raise or lower your POI until you are. Then move to 1 and 5 and adjust again if necessary. The idea is to find what Phil Kiner calls your natural POI. I'll also add the following: you are unlikely to get it perfect in one session. You'll be close, but next week you may be just a little off. That's fine. Make a minor adjustment and go with the flow. Sooner or later your subconscious will tell you "this is right".
Now go back to the target board and do the horizontal line bit again. You are shooting to determine your actual POI for reference. Just for the sake of argument, let's say you find the gun shoots 4" high at 13 yards. Write it down. That way you have a reference you can always return to if you have been experimenting, or you change guns, etc. The core idea here is you made the gun "shoot where you look" by actually firing at real clay targets, and only then determined your actual POI for the record. It is sheer lunacy to arbitrarily say I will set the gun up for x" high or x/y percentage, and then go shoot.
Now for patterning. Patterning takes a lot of time and effort if you want good, believable results. If you have a "good" barrel, almost anything you feed it will work well. I'm well past thinking another 10 pellets in the inner 20" circle is going to get me another bird. Personally, I'm much more interested in uniformity than I am in actual PE (Pattern Efficiency). If you have a "bad" or finicky barrel or choke, you'll be testing many more loads.
Here's the deal. Trap is shot with tight chokes. Tight choke forgive a multitude of sins, patternwise. So I'll suggest you take your favorite singles load and shoot it for singles. Now that you have your gun set up properly, you'll know if the load is performing. Do the same with caps. Good loads are the ones that work. You'll quickly find a few that you really like.
If you are still interested in patterning, or simply want to find out which of three pet loads works best in your gun, go at it. I do all my "official" patterning at a measured 40 yards. I do that because I shoot tight chokes and you simply cannot tell anything of real value at 32 or 34 or 35 yards unless you are shooting a modified or looser choke. I also do all of my "serious" patterning in the winter. There are two reasons for this. First, as Bob Brister notes in his book, patterns from a full choke spread in a trumpet shaped pattern. They start out tight, stay together for quite a while, then suddenly begin to flair out much more quickly, just as a trumpet does near the front. The tighter the choke, the farther away this happens. Second, air is denser in the winter and causes patterns to disburse more quickly. I want to magnify differences, and patterning at 40 yards on a cold winter day is the way to do that.
Remember, right now I'm comparing two loads that I already know work well. I'm not interested in determining whether I got 68% or 70.4% PE. I'm interested in comparing the performance of two loads, one against the other. The one that gives me the most even patterns wins. I shoot a minimum of ten patterns each, often more, before I form any kind of conclusion. I shoot from a bench, carefully, and use Shotgun Insight to analyze the patterns. It tells me what I need to know.
BTW, if you find a load that gives you 70% PE @ 40 yards when the temp is 30 degrees, that load will give you 82+% PE in the warm summer months. That's why I pattern in winter. Small differences are easier to pick out.