The main character of Red Dead Redemption 2 doesn't have to use the same horse throughout the entire game. You can change your horse to one with better stats (i.e. more health points). On this page, you can learn where you can find the best horses in RDR2. The first recommended horse can be purchased at a stable while the second animal can be found in the game's world.
The best horse you can purchase
Wait with getting this horse until you reach the fourth chapter. The option to get new mounts is tied with your progress in the main storyline.
Reach Saint Denis and visit the stable located on the outskirts.
The stable should have a black Arabian horse in its offer. The default price for this horse is $1,050 but having a lot of honor points will give you a discount. The biggest discount you can get is 50%, but this is only possible when you reach the highest reputation level. It is more probable that you will get 15 or 25% discount.
The best wild horse
The earliest moment when you can start looking for this horse is after the prologue (when you start the second chapter). This is a great way of getting a good horse at an early stage of the game - thanks to that you will be able to build your bond faster.
Head to the partially frozen Lake Isabella. This place is in mountains, in the north-west part of the map. You can start your journey to that place from, i.e. Valentine. Also, don't forget to take winter clothes with you.
Once you reach the lake, you can either ride on the frozen water or on the shore. Look around a lot, and you should find the white Arabian horse presented in the picture above.
Remember that this is a wild horse. You can't approach it right away and mount. The animal is easy to get scared off. There are two ways of catching wild horses. The first method is to dismount your current horse. Lock the camera on the wild horse (L2 / LT). Start moving very slowly towards the animal, and remember to press Square / X regularly to calm the animal.
The second method is to use your lasso. Start chasing the animal. Use your lasso when you get close enough to the wild horse. After that, you have to dismount your horse and approach the wild one - press Square / X to calm it down.
It is highly probable that the wild horse will run away and disappear from your view. Luckily, you don't have to use the Eagle Eye. Simply follow its tracks left on the snow.
Mount the horse. You now have to complete a short mini-game during which you have to tame the animal - move the analog stick in the opposite direction to the horse's movements. Completing the game allows you to tame the animal. If you fail, Arthur will fall from the horse. This means that you will have to repeat all the steps.
Go to any stable. Move your saddle to the newly acquired horse - this will make it your main mount. The Arabian horse has very high "starting" stats.