Crash Course in Horses (for writers)
As a rider, trainer, and writer, I figured I'd post some helpful info aimed specifically at writers. lmk if it's helpful!
for the love of all that is holy, you do NOT flick the reins unless you are DRIVING A CART and not actually sitting on the horse. Flicking the reins shakes the metal bit in the horse's mouth, which they do not appreciate. To go forward, squeeze your calves or bump with your heels/tap with spurs. I'm so serious about this one. I don't let a kid ride until they understand that they NEVER shake the reins. It takes 2 seconds for someone to tell your inexperienced character not to do that.
The only reason people flick the reins with cart horses is so the rein taps their butt. The driving equipment is made so flicking the reins does not disturb the bit.
NEVER tie a horse up using the reins on their bridle. That's a fast track to broken tack at the best and broken jaws at the worst. For long rides, always keep a halter on underneath the bridle and either carry a lead rope with you or keep it attached to the halter as you ride and wrap the end of the rope around your saddle horn. You've got something to tie them up with and something to hold onto if you dismount.
Saddles aren't* interchangeable. Any good horseman will make sure a saddle fits a horse properly. A saddle may fit multiple, similarly built horses, but you can't just chuck a saddle on another horse and go. *A good tactic for your long travels, especially in historical/medieval/horse using times: rider has two similarly built horses and 1 saddle that fits both of them properly. Rider rides one and leads the other and then switches off as needed.
Horses eat a LOT. They need to eat every 4 hours minimum to keep their digestive system working properly and the average thousand pound horse (500kg for metric) eats 20 pounds of forage (hay/pasture grass) a DAY. If horses don't get the amount of food they need, they can get ulcers and/or colic and die.
Your characters should take grooming supplies with them on their travels. A brush, a metal or rubber curry, and a hoof pick are sufficient to get the job done. Horses should be brushed before and after every ride and you need to pick their feet out regularly to avoid infections like thrush, especially if it's wet and muddy.
Hoof maintenance is non-negotiable. No hoof, no horse. They need a trim every 6 weeks minimum. Horses who wear shoes need new shoes every time they get a trim. An experienced horseman might also be a farrier, but if you don't have one of those in your group your characters better drop in at the nearest village and find one.
A horse needs about an hour of REM sleep a day. They have to lay down to get REM sleep. The rest of the 3-8 hours of sleep they get is usually standing up.
If you're stopping for the night somewhere, find a couple sticks and string a rope or two between them to make a temporary fence instead of just tying your horse to a tree or whatever people do in the movies. Alternatively, string a line between two trees and tie the horses on the line.
Unexperienced riders are going to struggle to stay on at any gait faster than a walk. Athletic ability definitely helps, but it'll take them at least an hour or two to really adjust and have any hope of staying on at a trot or canter.
Different cultures have their own saddle designs and even to an experienced rider it's going to feel weird and their balance may be off until they get used to it. Unless you change saddle styles a lot (english to western IRL), it'll take a second.
Additionally, character height and proportions really matter. I'm 5' 2'' and I need my stirrups REALLY short. If I try mount a horse in a saddle that a 6' person just rode in, I'm going to 1) struggle to get up there and 2) I won't be able to ride nearly as well or nearly as fast unless I'm super experienced/balanced. Same is true in reverse.
In general, a bigger person needs a bigger horse. Their backs can only carry about 20-25 percent of their body weight safely so riders need to be assigned accordingly. There are smaller horse/pony breeds that can carry more than the average horse, and in the real world you might see a 6 foot cowboy riding a small horse, but that's not recommended for long-term health. Plus, a small horse usually needs a smaller saddle and it's not fun to ride in a saddle that's too small for your butt, no matter who you are.
Every horse rides differently. One may be super sensitive to pressure (spurs/heels, rein tension, rider balance or lack thereof) and one might need a LOT of encouragement to get going. Some horses are more forward (want to move and do it fast) than other horses. It takes a second to figure out what a horse likes and doesn't like. If your character is super inexperienced, they're going to STRUGGLE switching to an unfamiliar horse. If that horse is extremely well trained, that unexperienced rider is going to hit a lot of wrong buttons on accident.
Most male horses aren't stallions--even in ye olden times. Keeping stallions and mares together is a great way to have accident babies all the time and pregnant mares shouldn't be ridden past a certain point. Spaying a mare isn't a very common thing with today's technology unless it's medically necessary, so most male horses are castrated. Testosterone generally makes most male horses difficult to handle. Unless they're very well mannered and respectful of of other horses, they're going to be a gelding. No balls=no babies=no stress.
Riding a draft horse feels a lot like riding a couch that moves a LOT. They're bigger, so all the movements are bigger, which means it takes more muscle and balance to ride well. It's just physics.