Here is the full list of all 15 Horse and Go Seek achievements.
Complete the first level in Pleasant Isle
Unlock the Lazy River
Unlock the Desert Canyon
Unlock the Dark Woods
Unlock the Tranquil Taiga
Unlock the Tropical Islands
Unlock the Horizon Highlands
Find all of the hidden Spirits
Rescue all 6 Snapping Turtles in the Tropical Islands
Rescue all 6 Camels in the Desert Canyon
Find a Penguin in the Tranquil Taiga
Find a Cat in the Lazy River
Rescue all 6 Dogs in the Dark Woods
Complete all 28 normal levels
Complete the Bonus World