Skeet, trap, and sporting clays are three very different games.
I shot registered trap as an ATA member. I shoot skeet when I need a break or to loosen up.
Skeet consists of a high and low house throwing clays, and a half circle field, with 8 stations (9 total, 1, 2, 3, 4, 5, 6 ,7, 8 hi, 8 lo). A round is 25 birds. On stations 1 and 2 and also 6 and 7 you shoot two singles (1 hi, 1 lo) then a doubles pair (hi and lo thrown at same time). 3, 4, 5, and 8 is only one hi and one lo target thrown as singles. Usually, skeet guns have a field stock and short barrel. Skeet, Skeet 2, Improved Cylinder, and sometimes light modified chokes are used. In competition, there are different competitions for different gauge shot guns - 12, 20, 28, and .410 bore. 9, 8 1/2, and 8 shot is most common.
Trap is a little different, there are 5 stations in almost a 1/4 circle around the trap machine. A round of trap is also 25 birds, 5 from each station. The birds are thrown at an approximately 45 degree spread from the center of the machine. The trap machine oscillates left and right randomly - you don't know where the target will be going when you call. Registered birds fly at 42 mph, and climb - they are measured 10 yds from the house and are supposed to be 9 feet high at that distance. Regular trap is shot at 16 yards from the trap house - handicap is shot from 17-27 yards from the trap house. The further back you go, the more lead on the targets you need. In competition, you start at 20 yards handicap, and are assigned yardage based on how well you shoot. Generally, a 96/100 will get you 1/2 a yard. Doubles is another trap game - doubles targets are thrown at the 45 degree angle on the same path - you do know where doubles are going all the time. They come out together as a pair when you call - one right and one left. Trap usually uses a trap gun with a Monte Carlo or straight stock (so the gun "shoots high" and you see the bird break over the barrel) and full chokes are most common, though I prefer Improved Modified for singles (16 yds) and handicap. Modified can also be used for singles and caps, as well as the first shot in doubles. For doubles, depending on the gun I like a light modified first shot, improved modified second shot. 12 gauge is pretty standard - there are no special competitions for smaller gauges, but you can compete with 20 and smaller just as well. Usually, 8's are most common shot size - 7 1/2 is the largest allowed size for trap and are best for handicap (or for all games in the winter) and singles. 8's or 8 1/2's work good for 16 yard singles. I like 9's for the first shot in doubles.
Sporting clays I have not done yet but I hear is a lot of fun. Usually uses a field gun, although I know a lot of people using trap guns for sporting clays. I think open chokes (IC, Skeet, etc.) are common with modified being tightest you want to go. SC targets can come towards you, go over your head from behind, come across, fly at all angles, even go straight up in the air. There is also a "rabbit" target that bounces across the ground.
For all games, nothing beats a 1 1/8 light target load of 8's. Thats my old standby, either Winchester AA's or Remington STS's.
Be Safe! Have Fun!
Registered Trap shooter, shotgunner.