Every horse has their quirks. But regardless of the horse, there are rules you should always follow to get the results you want:
- Make sure your horse is paying attention to you from the very beginning
- Use the lightest cues possible, and do not bump around.
- Make sure your position is proper and your horse is balanced.
- Give and take; give your horse space to do his work but at the same time make him respect you.
- DO NOT EVER let him get away with anything. Make him give you what you want.
Now, take a look at the rules and see how they fit with this particular lesson horse. To me, it sounds like this horse is a little skittish, has a lot of pent up energy, and is unfocused. So, starting at rule #1: make sure he's paying attention. From the beginning, even before you get into the saddle, form a bond. Greet him like a horse. Horses greet eachother by lightly blowing into eachother's nostrils. So, hold your hand out and gently blow on his, and wait until he blows back.
As you tack up, talk to him quietly. Be gentle and easy; don't hold his hooves up too long or thump the saddle onto his back. When you lead him to the arena, make him walk beside you and not in front or behind. Make him stand quietly as you tighten your girth and adjust your stirrups and get on. Get someone to hold him if you have to. Then, as you warm him up at a walk, speak to him quietly. Gently play with the reins and make sure, in the least, one of his ears are flicked back and paying attention to you. As soon as both his ears go forwards, you know he's paying attention to something else. This means that if someone else clucks to their horse, he'll be paying attention to that cue instead of yours.
Here's where rule #2 comes in: use the lightest cues possible, but use all that are nescesary. Don't kick him or yank at the reins; a simple gentle squeeze or half halt are what to begin with. This also increases his focus on you. However, make sure to have a steady but light rein contact at all time. And it should be very consistant. Soft hands. Don't pull on his mouth.
Rule #3: your position and his balance. Remember your 3-point : heel, hip, shoulder. Don't let your legs slide forward or back. Sit on your seat bones. Look up. The basics are everything. Once you know you have your position down pat, think of his balance. He moves very forward, no? This probably means that he's throwing a lot of weight on his forehand. You can fix this by backing him up a few steps. You should feel his back rise up a bit as he evens out his weight and 'gives you his back'. This allows for better control and a smoother ride.
Pick up a trot. Make sure you are posting low in the saddle and not thumping on his back. Keep your hands and legs quiet, but going back to rule #2, use the cues you need. Don't expect him to do the work for you. Rounding a corner: tighten the inside rein and apply a slight amount of inside leg. Moving down the long wall of the arena: half halt halfway through to keep him at an even pace. Moving down the short wall: use your inside leg to push him to the outside. Do lot's of bending to keep his mind off of speed.
If he starts to get fast at the trot, don't just pull on his mouth. Instead, use your seat to control the pace. If you want him to go slower, post slower. Use small half-halts and sit back, using half halts in the corners mostly. If he starts to get TOO fast, bring him into a circle and focus on bending until he evens out the pace.
When you pick up a canter, here's where it's especially important to think of rule #4: give and take. Back him up before you begin so that his weight is still evened out. Then, pick up a trot. DO NOT ask for a canter until you have established an even trot. Then, in the corner, move your outide leg back and squeeze. It should be a smooth transition, not a run into a canter. Sit in a deep full seat and allow him to move freely and evenly. Do not restrict him by leaning over his shoulders. However, at the same time, keep an even pace.
The same rules apply as for the trot as for the canter. Bend, bend, bend. Lot's of bending. Circles, serpentines, figure 8's... anything to keep him bending. Half halts in the corners, sit back, and use light and quiet aids. If he gets really speedy, pull him into a circle, spiralling smaller and smaller until his pace has reduced to what you want. Don't let him cut his corners, either. When he goes fast, it's harder for him to stay in his corners. Force him into them. Force him to bend.
And rule #5... well, that's pretty self explanitory. Make him respect you. Don't let him get away with anything.
Anyways, sorry for the incredibly long post. I hope I helped, and good luck.