First, you need to decide which hand to hit the ball with. Hold the ball in the opposite hand. Make sure you keep the ball steady.
If you can't palm the ball, then just hold the ball until you're about to hit it and then turn the ball so your hand can strike it.
The most important part of the underhand serve is to make sure you hit the ball with the bottom side of your fist (as shown). Make sure to place your thumb on the outside of your fist.
Notice that even on the backswing, the ball is held steady in my off-hand.
You don't need to reach back far, the force needs to come from the upswing of your arm. The ball is still steady!
You want to make flush contact with your fist onto the ball.
Following through ensures that the ball will go where you aim it and adds a little more power to the serve.