Skeet, trap, and sporting clays all use those little orange clay targets, but the differences between the games is how those targets are presented. Skeet is shot by moving around a semicircle, and the targets come out of a high house and a low house. They fly the same path every single time, crossing over a stake in the middle of the field. In addition to shooting single targets from the high and low houses on all stations, doubles (one target from each house at the same time) are also shot on stations 1, 2, 6, and 7. Moving around the semicircle from station to station gives you a variety of shots to make, from overhead to quartering to crossers, even though the clay targets fly the same way every time they leave the house. Because you will be shooting doubles on some of the stations, your shotgun needs to be able to hold two shells...in other words, that means an over/under, autoloader, or pump (least preferrable of the three), but no single-shot jobs. Use nine shot target loads and skeet chokes if possible, they will give you the best combination for a good pattern, but 8 shot and/or an improved cylinder choke will work well also. Anything larger than 8 shot or tighter than an IC choke is going to be detrimental.
Trap, except for a specialized event known as doubles, always involves a single target being thrown from a trap house situated 16 yards in front of you. The house conceals the machine from your sight as it moves left and right at random, so you don't know which direction the target is going to fly, except that it will go somewhere away from you. Because you're shooting from a greater distance, tighten the chokes and up the shot size...I always shoot with 8 or 7 1/2 shot and a full choke, but different combinations work for different people. Trap guns only need to hold one shell, and they're instantly recognizable for their long barrels and high ribs (the guns technically shoot a bit high, so that you can hold a bit lower and see the target while breaking it, instead of having to cover the target with the muzzle and lose sight of it).
Sporting clays is probably the most fun, and unfortunately, the most expensive. It's like golf in that you have a course to follow, but instead of playing holes, you go from station to station and break the targets that are thrown to you. Sporting clays is meant to mimic game shots, so the target presentations can vary immensely, and I've seen some people get really creative. The targets also vary themselves: in addition to the normal skeet and trap clay pigeons, sporting clays courses may include "midi" targets (about 3/4 of normal size) and "mini" targets (less than 1/2 normal size...we call them hummingbirds here!). There are also "rabbit" targets that bounce along the ground, and "battue" clays that are thin and fly like a bladed frisbee; it's not meant to mimic any real hunting shot, but just to be a challenge. Even the colors of the targets can change; at our local course here in Maine, orange, black, and white (yes, white...we use white rabbit targets on occasion to replicate a "snow bunny"!) targets might all show up on any given day. The fun of sporting clays is, it's so variable from place to place, and some people really know how to use a course's natural features to put together a fun round. Good setter-uppers will have the targets curling around trees, bouncing behind bushes, and might even add in a few features that would definitely not be found in nature (the rabbit target that ran up a skateboard ramp this summer was a good example!). Another nice feature about sporting clays is that if it's not in ya to actually hunt game (which goes for me as far as birds are concerned...no I'm not a pansy, it's just that I've spend an entire lifetime in ornithology and couldn't really bring myself to kill a bird even if it's delicious!), you can still "hunt" the clays as if they were the real thing.
With regard to your question re: which gun to use, I've seen pleny of guys using pumps and semiautos for skeet, trap, and sporting clays. Break-open guns are preferred in the Olympics and at high levels of competition elsewhere, but I've been beaten plenty of times by someone using a semiauto while I shot my break-open trap gun. It comes down to what works and fits best for you and your physical makeup...some guns just don't fit certain people. If you're on a skeet field, you're shooting skeet, regardless of the gun you're shooting! In fact, there's no need to spend big bucks on a super nice break-open gun unless you're sure it's a sport you want to continue in and invest some money in. Yes, I have a break-open for trap, but that was only after a very enjoyable year of shooting trap with a semiauto and deciding that a break-open BT99 would be money well spent (and even then, I bought it used!).
