First off, my gun is not cylinder bore. I have short screw in choke tubes. In my gun, the full choke tube is pretty much useless since the other two will both pattern better than it does with the system I use.
These are my thoughts. They are not carved in stone, but they work for me. Any critisism or discussion of another system or idea is just my thoughts. Not arguing with how anyone else does things in any way. This is what works for me and what I see or have seen.
I don't use 777 in the shotgun. I use Pyrodex or real black. I have used 3f, but the gun patterns better with 2f. I suspect that the 777 is too fast to produce best patterns. 2f kicks less than 3f and patterns better. Unless you have ignition problems, that is pretty much all there is to say on powders.
I use a thick NitroCard wad that is right at 6mm thick, or pretty close to 1/4th. I don't nick it or drill holes in it. I load thru the choke.
I mostly do not use a cushion/lube wad at all. I load the shot right on top of the thick over powder card. If I do use a cushion/lube wad, I put a thin over shot wad down on top of it before adding the shot. With Pyrodex RS, I can shoot more than ten times and never swab. For some reason the cushion wad helps patterns when you shoot the very small shot sizes in my gun. Out hunting, the most I ever have to do is run a lubed patch down the bore after it is loaded a couple of times a day. With black I have to swab a little more often, but not much. I suspect that the heavy nitro card scrapes most fouling out as you load and shoot. On damp days, the gun needs to be swabbed well between shots if it may be a while before it gets fired again.
I use the thin commercial over shot cards. I have never had a problem with them in a single barrel gun. They are suspect for use in doubles because the shot load can creep under recoil from the other barrel. If you shoot a double, you need to check that amount of creep in both barrels by shooting the other barrel with the gun held loosely. Then check to make sure the load has shifted in the other barrel. It is just part of learning your gun, but it deserves mention. None of them move if you hold them tight to your shoulder and absorb the recoil usually. You have to let the gun jump when you fire it to check for movement. No shift is ok. Even a little becomes a problem if you shoot the right barrel 4 times and the left load shifts a little each time. You really should check the loaded barrel when you reload the other barrel every time because of this. This is also where nicking the edges shows it's problems. Part of what keeps that thin over shot card in place properly is the air seal creating a vaccum state if the card attempts to move foreward. If you nick them or poke holes in them, the load loses part of what held it still in the barrel in the first place.
My current gun likes a 3:4 loading best. That is the very top range/pattern compromise loading. It is hot enough to do business at 40 yards and patterns well enough to do it. 60:80 of number sixes is a pleasant load to shoot and it will handle most of what guns like mine do well. 90:120 hurts to shoot, but the turkeys hate it worse than I do!
I bought a shotsnake with three settings on how much shot it will throw. I can play at the lowest setting with 50 grains of powder, or go to the highest setting with 80 under it and do just about anything that the small light single barrel guns are good for.
Some guns show a marked preference for certain shot sizes. I can show you a 410 that is ok with sixes, but will completely destroy a rabbit at 30 yards using 5's. You have to pull off if you want to eat the rabbits you kill. You can pattern that gun with shot size. I have seen several such guns. If you are shooting smaller than 20 guage, you will often find that this is true for your gun.
Shotcups are a completely different thing altogether, but need to be discussed. CVA makes a shotcup that has no side splits. If you leave it that way, it makes a good 30 yard prefragmented slug, but isn't much good for anything else. It takes considerable experimenting, but once yoiu figure out how far to cut the sides to suit your gun, you can shoot pretty good patterns at longer ranges. They just plain are a waste for most purposes, but they shine for that little bit you might need for turkeys and such because they let you run a little hotter on the powder and keep your pattern. Standard wads are hit and miss like any other loading component. I use a thick nitro wad under the plastic wads, so they really don't benefit me enough to bother. I am experimenting with plastic cups for use with steel shot in barrels not designed for steel. They are not cut down the sides at all and are designed for modern magnum loads using steel shot. I am cutting wings on the bottom bevel to slow the cup down so it separates from the shot quicker. I can get a good 35 yard pattern, but it is still to tight closer in. The other use for modern wads is the Lyman slug designed to be loaded in a modern trap wad. I have screw in chokes and I am not supposed to shoot roundball thru them. The Lyman slug in a modern wad is supposed to be ok? Anyway, that option it there also. If I can get the ducks to cross at 30 yards this year, I am going to be eating duck!
There are volumes more to type, and there are several other folks here using systems that work for them. Have patience at the target board and use big sheets of paper so you can see what the various changes are doing. Most of all, slow down. 50 shots at the board is supposed to be something you were priviledged to be able to participate in! It isn't an imposition to have to shoot another 50 from where I stand! I know little about the target games, but I would expect that those folks shoot that much in a day. I had to bend my barrel three times and experiment for a while with this gun. I had a 20 I never did figure out! Wish I had it back with what I know now! Just slow down untuil you find what that one will and will not do. I am sure others will add more!