Having walked for a good amount of my life, and having set time aside when I was young to practice the trade, I am at such a level where I may be able to pass on some basic sauntering maneuvers. I am not an expert, but I am by no means a novice.
I find the first thing that absolutely must be done in order to start a true saunter, is to clear your mind of the illusion of your need to accomplish anything. This can be hard to do, so I suggest you first study in the area of Realizing How Unimportant Things Are.
Once your mind is clear of that pesky to-do list and you have literally nothing to do aside from exist, take a few steps. Keep your body relaxed and let your arms sway however they do naturally. No need to have a brisk pace, but don't go too slowly, or else you will distract yourself by concentrating too much on where your feet currently are. While it is important to know where your feet are, you shouldn't be bogged down by it.
The next key thing to do is to disconnect your mind's conscious control of your legs, and to think of nothing of particular importance. Much like your heartbeat, breathing, or blinking, your natural walk follows a pattern. If you are old enough, your walking pattern has been subconsciously imprinted into your mind so deeply, that you can walk without thinking about it. This will be hard for beginners, due to your want to learn how to saunter, it may come up in your thoughts every now and then. Do not let it.
Once you've cleared your mind and set your walk to auto-pilot, let your thoughts drift. You'll know your sauntering by the look that others give you. It will be a look comprised of disbelief in the fact that you are indeed so calm, disgust in the fact that you flaunt your calm so publicly, and the slightest hint of jealousy over your obvious freedom.
Beginners Beware! It can be easy to become lost in a saunter if you do not set into your mind certain "alarms". Perhaps use your internal clock to know when you have sauntered for long enough, or saunter until an outer event such as your ringtone or a car horn stops your saunter. Sauntering for too long is dangerous to various versions of your health. (Physical, Mental, Social, Emotional, Financial). Saunter Safely!
Below is a picture of me with my good friends in mid-stride of a synchronized saunter.
Thanks for reading!