The saddle club is a book series iirc. I just remember the TV show being popular when I was a kid, but I've never been a horse person so I can't say if the books or show are worth it.
It was kind of like The Babysitters Club. Fun and worth the read if you're nine, but honestly someone coming into it as an adult probably won't like those books.
Actually, this is me sharpening my genre whetstone. I'm writing a book set in a magic school, main character is taking a pegasus-riding class, and its becoming clear that in the climax of the book I'm going to have to describe some very complex aerial horsemanship when I barely understand how a horse thinks.
I'll have to level with you, horse fiction doesn't usually accurately portray the way horses think. If you're going with fictional magical pegasuses I wouldn't worry about being accurate to real life and just do whatever works for you.