I spent five years working in the pet industry. I've shared my home with every kind of pet you can pretty much get. I've had horses, rodents, fish, birds, reptiles, amphibians, insects, arachnids, cavies (you know what a cavy is right?...read on!) marsupials, and various ordinary and not so ordinary mammals. I'm not a veterinarian, but I put her kids through college back in the horse years. Trust me.
I'm going to start with horses, because I think not only are they a source of massive literary inaccuracy, they're also an untapped gold mine of plot twists. If you plan to put horses in, I suggest either owing one, borrowing one or, even better, signing up for riding lessons. Make sure you pick one of those sagging barns with the muddy pastures that likes to give the "full horse experience" though.
That usually means you'll be mucking stalls, dragging feed buckets around, picking hooves and doing a good deal o the work required to really get to know horses.
So aside from glaring technical inaccuracies like putting a "martingale" on the horse's head. (No. Nonono) the biggest problem I see with horses in fiction is that they are often metamorphosed into big, somewhat useful, dogs. So very not right.
A horse is not a cuddly puppy. Even a very well trained horse. Even Tonto's horse. (Dont. Even.) If a horse decides it wants to cuddle you, and yes, sometimes they do, you will most likely end up on your ass. Or hanging like a sloth from the horse's neck, the stall door, tack rack or whatever you managed to snatch before Scout there tried to knock you over. Horses are BIG, really big, heavy and strong. They do what we tell them only through the subtle manipulation of cues, treats, threats, cussing and sometimes tears.
This is good! This is plot fodder. If your hero is riding or living around a horse for any length of time, the odds are it will try to kill him at least once. Even nice horses spend their idle hours dreaming up ways to kill their favorite person...or themselves. They do that, too. If there is something deadly poisonous within reach your horse WILL eat it. A lot.
They will also: bite you, step on you (and refuse to get off, and act shocked when you push on them) kick you, spit on you, roll on you (Sometimes while you're in the saddle--how about during that big getaway scene?) poop at the worst possible moments, stop for a five minute pee (again, during that getaway?) stumble, trip (on purpose, I swear to god) swerve at random moments, and jump off cliffs willingly if a bush looks at them wrong.
A horse is an unpredictable element. And that is how it should be. Right? This is not a well trained circus gazelle. Horses fart, and waddle, and go into heat, and run away! Oh how they love to run away. If your hero doesn't tie that knot, just right....horse gone. Or better, horse tangled into a three point yoga pose with leadline twisted around each leg and its throat (even though its not long enough to do that) and choking itself while thrashing to be free.
Horses. Done right, a plot goldmine.