Great shots! As an artist and novice photographer myself, I will take liberty to mention several points to help out new photographers that have really helpd me...
Especially with animals..get on their level. As a matter of fact, it is often more interesting to photograph from many levels, at eye level is recommended..but going below the animal's eye level makes them appear larger, giving the appearance of importance/ strength, while photographing from above can do the opposite.
With most portrait photography person or animal...fill up the lens with as much of your subject as possible. Backgrounds are nice but usually not as nice as your subject!
A single lens reflex camera (with interchangeable lens) is great for blurring out everything but your subject, but most of us own point and shoots. Good thing is many have "MODES." Use them! For example, taking a pic of your horse's muzzle, use protraiture mode(which will tend to blur out everything but your subject), or if you aer photographing a horse in motion use action mode (which will cature faster mevement without as much blur).
And my fave!!! Don't always center your subject! There is a rule called "the rule of thirds" where you put your subject in the top third, bottom third, right third, or left third of your shot as opposed to the middle. Makes a more interesting shot.
And remember there is ALWAYs an exception to every rule!