To start training one needs to own a tamed mount (donkey's can be obtained via quests early on) and ride it. During riding a mount or driving a vehicle pulled by a mount one owns (!) the mount levels slowly up and with every level it makes or every successful taming even the training skill of the player increases too.
There are special clothes that increase training stats of a player avatar sold by luxury vendors.
The training skill might affect how likely it is to be thrown off an animal when training its skills.
Once a player avatar has reached a training skill of Beginner: 5, he can try to tame wild horses.