There are two common ways to hold a sling, and there are two common ways to release the stone. You'll see what is meant in a moment.
HOLD 1: Loop cord is wrapped around the wrist with cord falling through space between thumb and index finger on palm side of hand. Index finger and thumb grasp the knot on the RELEASE cord. At this point, the ANCHOR cord should be enclosed in the circle formed by the thumb pinched to the index finger.
HOLD 2: Loop cord is around middle finger with length of cord away from hand. The knot of the RELEASE cord is held between the thumb and index finger.
LOADING: The easiest way to load a sling is by correctly holding the cords in one hand and let the pouch fall suspended. While the pouch is suspended, load and center the projectile in the pouch.
RELEASE 1 (Underhanded): After loading the sling as described above, swing the sling FORWARD as in fast-pitch softball. Your arm should be moving at the shoulder, not at the wrist-- the cords are an extension of your arm, and you are making a large, somewhat diagonal circle in the air. Practice releasing your thumb at the moment when the projectile will leave at a 45 degree angle. You can even get a running start with this method. This type of swing is great for power and distance--lobbing. It lacks accuracy though.
RELEASE 2 (Over the Head): This method starts as the first-- underhanded and forward. But, as the sling is swirling, bring it over your head, your arm turning more at the elbow than the shoulder. The circle over your head should be about perpendicular with your body. Again the sling cords are extensions of your arm. This time, however, you throw like a baseball. Throw your arm like you would if you were throwing a baseball but only when the pouch is swinging forward--add the strength of your arm to its momentum. Practice to get a feel for the release point. You'll feel less power with this method, but the accuracy will be up.
SAFETY: It should be remembered that slings, in the hands of practiced men, have been used to hunt large beasts and hurt men. Slings in the hands of untrained men are able to break windows, dent cars, etc-- PLEASE be careful and abide by all laws. Stay away from the private property of others and don't ever aim in the direction of another person. Slinging, hurling, lobbing-- it's all fun, but it is very challenging. Don't give up. With practice, you'll be throwing very well.