Before we're getting into horses you might wonder how to tame a horse, right?
If you do know how to tame a horse skip this part, I'll just add this here so it won't be asked later :)
Before starting your horse search you'll need to buy carrots, carrots can be found in one of the markets in town. You only need 1 carrot for one horse and carrots cost 350 coins each.
When you get your carrots start searching the horse you want to find, when you find your dreamhorse, open your inventory (I) and take your carrot to your characters hand, then go close to the wild horse and click it, boom it's now yours! Easy peasy lemons squeezy.

In this map you can see location areas circled where the horses usually are.
(forgive me if some horses have wrong "color" or breed name.. I don't know them in english that well)

I know there might be few horses missing in map, so if you find them before I do feel free to leave hints where to find them in comments and i'll go check it out!