One of the things I love about the participatory nature of the Church is that we are given the opportunity at a young age to learn public speaking by giving talks in Primary. (Oh, you thought this was going to be about sacrament meeting talks? Nope.) I think learning to give a talk in Primary can be one of the most beneficial aspects of the Primary program. Some kids are uninhibited: they'll march right up to the podium, yank the microphone halfway down their throat, and belt out a talk. Others will freeze unless prompted by their parents -- and in some cases, the parents end up giving the talk for them.
We have five young boys; two of them have given talks in Primary. One of them was very shy about speaking in front of people, but we used a great method to prepare him to give his talk and he got right up and got through it on his own. My mother came up with this method when I was in Primary, and I think it will work for most young children. Here's how it works:
Fold a piece of unlined paper in half so that the short edges of the paper meet. Fold in half again in the same direction. Now fold in half the other way. When you unfold the paper, you will have eight squares (okay, they're really rectangles, but close enough) defined by the folds. (For older children, you can fold it so you end up with 12 or 16 squares.) Sitting with the child, talk about what she wants to say and draw a picture in each square to represent that part of the talk. It shouldn't be anything fancy, just one or two iconic images that can help the child remember that part of the talk. Make them big enough to fill up most of the square. (For older children who are beginning to read, you might add a word or three.)
Now go through the talk with the child several times, pointing at the pictures as you go along. Have the child repeat what you say. Then have her do it on her own, with help from you as she needs it. It's amazing how quickly children pick up on this, and how much they will remember. It is not essential that she memorize it word for word (though this works better for some children); the key is that she knows the concepts associated with each picture and can express them. Then have her practice several times in front of the family. Set up a music stand or a stool or something as a stand-in for the podium, and have the family sit and listen (Family Home Evening is a great time to do this).
On the day of the talk, go over it a few times right before leaving for church (and if Primary is after sacrament meeting, maybe right before the end of sacrament meeting). Go to the Primary sharing time where she will be giving the talk and stand in the back of the room, where she can see you. You can silently prompt from there if she gets stuck, and she will be reassured by your presence and praise, but will be on her own in front of the group.
I think most children will succeed with this approach, and they will learn some great skills: memorization, public speaking, outlining; and they will develop greater self-confidence. Hey, it worked for me: I love to give talks!