Here is the full list of all 9 Pen Pals achievements.
Collect your first medal for a job well done
Collect a silver medal or better on 5 stages
Collect a gold medal on 10 stages
Discover a horse
Against all odds, get two horses to meet
Earn a score of 6,000 or better. Go on, keep hyping six.
Have a single animal transform 4 times in one drop action
Have a single animal transform 7 times in one drop action
Have a single animal transform 10 times in one drop action