Take your pictures quickly. The longer you hesitate, the better the chance that your subject will move away, break their pose, or become tired and stop smiling. Start taking shots as quickly as you can, and don't worry about getting the camera perfect before the first shot. The faster you shoot and the more photos you take, the better your chances are of getting a good one.
To avoid red eyes in your photographs, make sure that your camera either has built in red eye reduction or you change the direction of your flash. If you did shoot a photo and it contains red eyes, you can easily remove them through the use of a graphics software such as Corel or Photoshop.