While previous Zelda games have tended to hand Link his steed as part of story progression, Breath of the Wild is once again different. With its open world and open attitude to progression, you can grab a horse pretty much as soon as you leave the Great Plateau at the end of the game's opening: but you need to know how.
Horses in Breath of the Wild can be a little temperamental, and not all horses are made equal. The system isn't explained in much detail in-game, but if you'd rather not just naturally discover the mechanics through trial and error, allow us to step in and explain...
Wild Horse Taming in Breath of the Wild
The first thing you need to do is find a wild horse. This shouldn't be difficult, to be honest - they're almost everywher eout in the overworld of Hyrule. Anywhere that you can see a wide open and relatively flat area there'll be horses. Sometimes there'll even be wild horses being ridden by enemies, and if you knock those enemies off you can claim the horse.
The first place the game prods you to in order to find a horse is the Duelling Peaks stable - it's on the way to Kakariko Village when you're following the main story. That's your first stable, and right near that stable there are horses - and even a side quest to teach you how to catch horses.
How to Catch and Tame a Horse
When you see a herd of wild horses, single out the one you want from afar first. Get a color and pattern you like. When you're ready, crouch and begin to sneak up on the horses.
If you have any stealth-enhancing elixir, food or armor, feel free to use them to make matters easier. When you get close to a horse, press A and hop on its back.
Here's where it gets a little more complicated: all horses in Breath of the Wild have statistics in the background and a temperament. Depending on these stats and the temperament some horses are easier to break in than others.
While the horse bucks and goes wild, you have to keep soothing it with L to calm it down. Every moment you're on it while it's bucking costs stamina, however, and if you don't soothe the horse successfully in time, you'll come tumbling off and have to start again.
Here's the key: higher-end horses will most often require a lot more soothing to tame, which means you'll need more stamina. You can get more stamina by completing shrines or by creating stamina-boosting elixirs and food - this will help you to break in a better horse.
Better quality horses will have more stirrup icons when you're riding them, or be faster or more obedient. Speaking of obedience...
How to Keep & Train a Horse
Once you're on the back of a wild horse your next step should be to take it to the nearest stable, which will be marked on your map if you've been there before. Stables are all over Hyrule, so you'll never be far from one.
Once you reach a stable, remain on the horse's back and draw up close to the stable front desk. Chat to the chap there and he'll let you name the horse, officially register it and kit it out with some gear for you to ride it more properly. Once a horse is registered you'll be able to see its stats in full, too.
Once that's done, the key thing is simply to keep using your horse. Horses are generally disobedient early on, but as you build your bond with them they'll be more likely to behave themselves.
Increase your horse's bond by soothing them any time they do wrong - if they refuse a command or begin to veer off in a different direction to what you want, soothe the horse with L to get its mind back in the game. Over time and use your horse will become more obedient and be easier to control as a result.
If you want a fully-trained horse without any of that hassle, look no further than amiibo. Scanning a Link from the Super Smash Bros. range of amiibo will give you a special one-time reward - classic Zelda horse Epona with amazing stats and with her bond maxed out already.