Foals can gain skill based on their parents. If the mother had a high racing/jumping skill the foal may get it as well. I don't believe you running the foal around did anything towards training, it was genetics.
You train racing skill by running around or on a racing track and jumping skill with cavaletti's (then later jumps) but only as adult horses. Foals can learn things like being obedient, or fast, etc. So "traits" not "skill points"