I tried cantering on a serpentine for the first time in my lesson today.  I was supposed to change leads (I don't counter canter) with every curve.  However, I got pretty discombobulated (occasionally signaling for the wrong lead)--also my horse kept breaking into a trot before I got to the other side of the arena.  Any suggestions so I don't disgrace myself again?

Canter on a figure-eight and change at the center. The larger figure-eight will give you more time to think about what you're supposed to do. If you're not too sure of how and when to ask the horse for a change, it's not going to help to do it any faster by changing on a serpentine. Get really good at the figure-eights before you ask your horse to change more rapidly. Doing them on a serpentine before you're ready will set you and your horse up for failure.

 I don't have a problem going around the ring or with figure eights, but the twists and turns of the serpentine seemed to throw me off.   I guess I wasn't used to thinking that quickly--or I was thinking too much and confusing the horse.

Then try a three-loop serpentine, not one with more loops. In a dressage arena, a 3-loop serpentine is three 20m half-circle. If you're OK with a figure-eight, a 3-loop serpentine shouldn't be a problem. Try doing it on the ground first: mimic cantering on your own legs and think about when to ask for the change and how to do it. You'll be cantering, yourself, much slower than a horse, giving you more time to get coordinated.

Serpentine's are tough at first, but once you learn the geometry you'll figure it out.  I would practice the serpentine at the walk and trot making sure you have correct bend and remember, there are a few steps of straightness in the serpentine before you change the bend.  Your horse is breaking into trot because he's losing energy from behind.  Serpentines are tough for them too.  If you're confused and giving conflicting aids then I'm sure your balance is all over the place, which makes it even harder for your horse.

I would keep the serpentine at the trot and once you can consistently keep a nice forward pace at the trot with correct bend and are relaxed and confident, then move to canter.

Good Luck!

eta: placing a ground pole at each lead change might help you learn the geometry and give you something to aim for.

 Thanks @Boyle--yes, I think I was hesitating and that was why my horse was losing energy.  Also, @48--I think my instructor asked me to do 'too many loops' (at least for my brain) right off--a three-loop serpentine would have been much more managable.  I'll ask her next time if I can have a pole or an obstacle to help me make cleaner turns, too.

 I agree you should be on a 3 loop (in a standard arena).

 Something to try (to get the horse prepared for something new to happen when you cross the centerline (not at A & C, just when you change bend)) - exercises that do something extra at that poin for 3-5 strides. All of these will have him listening to you at the point of crossing the centerline:

• Trot the serpentine, but ask for 3-5 strides of walk as you change bend
• Occassionally, stay on the circle instead of changing bend (on all 3 cirlcles and in both directions)
  
• Walk or trot (maybe walk in warmup and later do the same in trot), asking for 3-5 strides of leg yield (toward the rail) after you change bend
• Canter, but ask for 3-5 stridews of trot or walk as you change bend

Believe it or not, the "simple change" in dressage is canter-walk-canter. If you haven't done it, or aren't good at flying changes, starting them on the diagonal is better than on a serpentine. Canter around the end of the arena, and start across the long diagonal. At (or just after) X ask for a trot (or walk), then pick up the other lead as you get to the next corner letter. This makes it obvious to the horse which lead you should be on - as opposed to the serpentine where, if you are doing it correctly, you ask for a chage while facing directly at the rail.