The trick to picking up your reins is a well schooled horse. Your horse needs to stay balanced on his hind legs and not come onto his forehand in the free walk. Then you put the buckle in your left hand, make a space for the reins in your right, pull up and grab where your reins need to be to move onto the next movement.
How do we get the horse so he is light and responsive enough to pull his neck up quickly and come back to the medium walk?
Perfect practice. Before you offer the free walk to the horse, develop a very good medium walk and then let the horse chew (read: not pull!) the reins down and continue to hold a contact. When the horse is consistently showing that they can stay connected in the free walk by holding the contact (however light) can we then give them the buckle and allow them to march on with no contact. Before the medium walk, we apply a half halt (so put our legs on - legs should not be on during the free walk!) and half halt again during the rein shortening stride(s) (it should take one stride by the time you have it "down").