As the manuals state, the training should start bareback. It will quickly develop Your muscles and balance. The saddle and the stirrup are for protecting the back of the horse and give support to combat moves. It is not good for a beginner to start with saddle and stirrup, because they will give a false sense of stability. As I wrote, the first excercise should be falling off the horse, to both side. Unexpected movements of the horse will occur, and falling should be a routine movement, not a rare disaster.
Horseback wrestling should be the next excercise, to develop your riding and balance further (and your falling routine, whether you will like it or not :) ). It has several variations, the basic round is between two horseman bareback. Who falls, loses. It can be played in groups, it is great for tactical skills. An endless and much more exciting form is the following: if a horseman falls off, he can get on any other horse except his previous one. This means that he has to rush an other horseman on foot, or help his teammate from the ground. A mounted warrior should react quickly if a footman grabs him, or falls off his horse.
Moreover the horses playing this game will fear nothing and get accustomed to „battle circumstances”.
After and only after acquiring such a good basic horsemanship should you take weapons in your hand. It is pathetic when someone calls himself mounted warrior but can do nothing if a footman grabs his wrist.