You can change your starter horse's appearance in exchange for Star Coins by going to your home stable, clicking on the noticeboard, and clicking on “Starter horse appearance.” Please note that you will need to sit on your starter horse to proceed with the change.
You will be given the same choices as for creating a new character, but the menu will look a little different. At the bottom right, you can choose to see your horse from different angles, zoomed in and out, with or without gear. You can also reset to how your starter horse looked before you started changing things, and next to that, you can choose to cancel if you don’t want to make any changes and click OK when you are happy with your horse’s new look.