There are 6 different skills available. A horse can learn up to 5 of them, each of which offers a different advantage in steeplechases and horse racing. Each skill can have up to 7 ranks. See more information on http://hhwa.wikia.com/wiki/Horses 

You can get/level up the horse skills by:

• Further leveling up your horses after lv10 gives them chances of learning skills by the spinning wheel
• Running steeplechase events
• Reaching new milestones