Find a horse in a plains, savanna, or similar biome
Make sure you have an empty hand
Right-click or tap the horse to mount it
Keep trying to ride it after it bucks you off
Wait for hearts to appear
Once hearts appear, the horse is tamed
Equip a saddle to control the horse
Equip horse armor if desired
Use a lead to move the horse if needed
Feed the horse to heal it or speed breeding