I can't help you with Showmanship since I just asked in another post what to do in Showmanship. But I can help with the others. You say western and that depends on if you are showing western pleasure or working western. Western pleasure is more flashy with their saddles and the rider sits pretty on the horse while you are asked to walk, jog, and lope both directions and to halt and rein back. Working western I've never shown in but from what I can tell it is based more on the horses ability to actually work on a ranch. The gates are still w/j/l but I believe it is more relaxed, and the tack allowed is also different.
English is more my subject of interest. If you are showing English Pleasure (called different things by different shows) than it is the same as western pleasure, but with a walk, trot, canter and of course English tack.
The keyhole race has already been covered by another member so I won't add anymore to that.
And trail is slightly different than what you would do on a trail ride. It is judged on successfully completing each obstacle that is laid out on a set course for you to ride/lead. I've seen many different obstacles but they are all designed to test your control and communication with your horse. The obstacles are generally designed to simulate what you might see or have to do on an actual trail ride. I have seen small bridges that have to be crossed, gates that have to be opened and closed, tarps, tires, and soooo many more things. It also depends on your level and how big the show is. If its a schooling show than there is not likely to be anything huge to worry about unless you have a spooky horse. Here is a video of a trail class at a top national Quarter Horse show just to give you an idea. Again if you are at a schooling show, most of the things wont be so hard.
(after the lady talks about her horse you see the horse compete)
Also a tip for finding out what goes on in a type of class.
I always youtube videos of that kind of class and watch what other people have done in that class. It gives you an idea. Hope this helps.