Not sure if you wanted to know about traits too or just skills? Some pet traits go hand in hand with the skills, some do not. For example, Hunting is a cat skill, but Hunter is a cat trait that helps that skill.
I am finding out that you can teach your pets some traits it not all of them. I have successfully taught horses fast, agile, and playful so far, and cats neat, adventurous, and non-destructive. The max is NOT three, that is only how many you can choose in create a sim, it is five. Untrained can be un-taught and replaced with something else.