I've shot many, many horse shows and made hundreds of horse portraits. In your shoes I would opt for a mirrorless system with a zoom lens that goes from slightly wide to slightly telephoto. These systems can provide outstanding image quality, are easy to use, are light and small and have interchangeable lenses. There is no question that the DSLR's have more complete systems and capabilities but I tend to think you don't need all that. Give it some thought.
I have to admit that I'm completely clueless regarding the mirrorless options. Any off-hand recommendations for where to start looking?
The mirrorless cameras don't have a swinging mirror and pentaprism. Instead of looking into a viewfinder you see the subject projected on a screen on the back of the camera, something like the camera in a smart phone. But they will outperform a smart phone by a bunch and have far more capabilities. Why not just go to Amazon and put mirrorless camera in the search block. You will find hundreds of them at prices ranging from around $200 to over $1500. Popular brands include Sony, Panasonic, Olympus, Fuji, Nikon etc. They aren't cheap but they are less expensive than DSLR's.