The list of objectives and the rewards you get from them:

1. Groom a Horse = Able to roll the die 3 times every 12 hours

2. Donate X amount of Equus to Ow = Able to teleport to the highlighted space

3. Give an apple to a horse that is 10 years old = 3 bonus rolls *horse must be exactly 10*

4. Enter X amount of competitions = Able to teleport to the highlighted space

5. Take a horse on a 5 hour ride = Move to a space of your choice within a radius of 4 spaces *Means take a horse on a 5-hour consecutive ride of any kind*

6. Find an Aging Point UFO = Able to teleport to the highlighted space

7. Remove a Pegasus' Wings = 5 bonus rolls

8. Win X amount of competitions = Able to teleport to the highlighted space

9. Open a Horn of Plenty = 4 bonus rolls

10. Donate X amount of Equus to Ow = Move to a space of your choice within a radius of 6 spaces

11. Give birth to twins = 5 bonus rolls

To clarify something about the rewards:

Number 1's reward is the only one to change how many times you can roll. It's the tutorial challenge that everyone will get.

2, 4, 6, and 8 allow you the option to move to a space the game chooses (apologies I don't know if there's any sort of pattern to the spot chosen)

5 and 10 allow you to move to any space within a 4 and 6 space radius from your current position respectively

All other rewards simply give you the option to throw the die again for however many times it allows.