Okay, let's give this a go!
Here are some topics where we've all talked about diagonals before: http://www.horseforum.com/viewtopic.php?t=13536 http://www.horseforum.com/viewtopic.php?t=11270
Basically, you know how when the horse trots, its legs move in diagonal pairs? If you're out of the saddle when the horse reaches forward with its inside hind leg, it's able to move more freely and so really use that leg to drive itself forwards or round a turn. I find that the easiest way to make sure I'm on the right diagonal is to concentrate on sitting as the outside shoulder moves back
. Somehow by remembering to sit, rather than trying to rise as the outside shoulder goes forward, I get it right more often!
When you canter, the horse moves its legs in a three-beat pattern, say right hind, left hind + right fore, left fore. The last leg is called the leading leg and the horse stretches this one out in front of it as it moves.
The foreleg closest to the camera is the lead in this image. You want the horse to be leading with the inside leg around a turn as it's more balanced and easier for the horse - 'counter canter' is a fancy dressage move. The easiest way to make sure that you're on the right lead is to ask for the canter on a turn, as the horse will almost always automatically go on the correct lead then. You can ask for a specific lead using your legs: if you want the inside lead, keep your inside leg in the neutral position or move it onto the girth a little, and move your outside leg backwards, encouraging the horse to take the first step on the outside hind and therefore lead with the inside fore.
There was a discussion about feeling your lead here
There are two types of changes: simple changes and flying changes. For a simple change, most people drop into trot for a few strides and ask for the correct lead going into canter again, but a correct simple change actually asks for a direct transition into walk, a stride of walk, and a direct transition into canter. A flying change is a much more difficult manouvre where the horse changes lead without breaking the gait. You should get your horse doing correct simple changes with a single stride of walk before you attempt flying changes.
On the lesson horses that I ride, on some of them you do indeed need really firm aids to get a transition out of them. While they're well-trained, I've been riding ones that are used to beginners who can only just walk and trot. Fortunately I'm moving up now and the horses are more responsive; it's downward transitions that are causing the problem! ^^
Anyway, consistent, firm reinforcement is what you need. Get the horse moving into a good marching walk first - your hips should move back and forth with his back, not side to side with his legs. Try asking with a squeeze first. If this doesn't work, immediately follow up with a kick. If this doesn't work, immediately follow with a firm squeeze and
your crop. You must reinforce every aid like this: if he doesn't speed up his walk when you ask you follow this pattern. You've just got to be consistent with him, and from the moment you get on be clear and firm with what you're asking. Do lots and lots and lots and lots and lots of transitions until you're absolutely incredibly sick of them, and it will make him more responsive. Reward correct transitions by releasing the pressure with your legs, but do not allow him to slow down or mess about with you.
Keep him at the speed and pace and rhythm you have asked for: when he keeps this up all the time without trying to break gait or slow down, then
you reward him by letting him go back to walk. Make sure your trot is not a slow lazy one but a good working trot. And make sure that when you ask for a walk it is the marching walk I described above. You mustn't let him get away with laziness, or he will, every time. Horses know how to minimise their workload! Make sure that you have a very quiet seat and aren't giving conflicting aids, e.g. pulling back while asking him to go forwards.