We've covered most of what you need to know about horses in our main equestrian guide, but the art of breaking and taming wild horses in Red Dead Redemption 2 deserves its own piece.
If you don't want to spend money on buying a horse and would rather avoid the legal implications of stealing one, the world of Red Dead Redemption 2 is full of wild horses you can break and tame.
Note that stables are only unlocked in Chapter 2. Any horse broken or tamed beforehand will be lost since you can't store them in a stable, and a Chapter 2 mission gives you a horse in a scripted interaction replacing whatever horse you may have tamed already.
Taming horses involves a mini-game of sorts with several stages and the difficulty of it depends on the "quality" of the horse. Elites are, naturally, harder to tame than standard horses.
When you've found a wild horse you want to tame, you can start the process by approaching it calmly and slowly.
Once you're close to the horse, you should use the focus button on it - Q on PC - to keep it in your sights. As you get closer, the contextual prompt for calming the horse will appear attached to X, Square or G. There is a white gauge on the right side of the screen, not the focus menu, which will decrease as the horse gets more and more agitated, and you can keep the gauge filled with the calming action. If it runs out, the horse will bolt.
When you're close enough, you can mount the horse with either Y , Triangle or E. Once you've mounted the horse comes the hard part. It will try to toss you off and you need to break it for it to accept you as rider. Basically, for a while you need to counter the horse's actions.
If it tries to go forward, lean back. If it wants to turn left, lean right. You control Arthur with the analog stick just like you do when walking. After a bit of struggling, the horse will be broken and can be ridden as your own.
This horse isn't your main horse yet, as you can only have one of those at any given time. To make a tamed horse your main mount, you need to saddle it.
All you need to is select the desired saddle from your inventory and equip it on the horse. Since wild horses are found in, you know, the wild, chances are you rode your previous horse to the location of the new one. If you don't want your previous mount to get lost, you can lasso it and lead it back to a stable - though generally they should just respawn in said stable on their own.