I agree that you need to go back to that barn and watch the instructor teach a more advanced lesson. If you are looking to further your training with your horse where you board then you need to make sure this instructor is suitable for you. You don't want to move and then find out their skills are lacking for your level.
As far as giving notice I can see both sides. Working in a barn is hard work, we all know that, and some barns rely on boarders to help get things done. I can understand that they need to find someone to fill your spot. However, this is what a barn manager does. They are responsible for the care of the horses and the up keep of the barn. They are to pick up the extra work when someone can no longer work their shifts until that person is replaced. This is true in any business. People quit all the time with out giving notice and things still get done. The only thing I would suggest, so you don't burn any bridges, is to tell them to find someone to replace you and in the mean time you'll only work once or twice a week there, or for 2 weeks, which ever comes first. This way you can walk away knowing you've done all you can for them.
This is what happens when you have democrats in office