Get closer to your subject when you can. Zoom your lens when you can't. Catch the emotion. Don't include things in your image that don't add to the emotion of the moment. This applies especially to portraits of people.
Discover the focus of your photo, then consider moving in even tighter. Are you taking a picture of a person? Consider taking a picture of only the upper body, or even the shoulders and head, or even just the face. Is a special gift being opened? Have the person pose, and catch the hands and the gift, and leave out everything else. Or catch just the face showing the emotion — many of the best photos catch the twinkle of an eye or the sly grin in a close-up, which are entirely lost if you include more than just the face.
Don't be afraid to move right up to your subject. Feel uncomfortable moving so close? Let the camera be the excuse. Or use the camera's zoom if necessary. But it's often best to move closer when you can. Your family will get used to your intrusion.
Avoid including distracting background details when setting up your composition. Move around your subject to get distracting poles, chairs, or people out of the picture, or crop them out by zooming in. Let the object of your photo tell its own story. Later, you may want to crop your final photo further using an image editing tool.
Although it's usually better to get too close rather than too far away, pay attention to things you want to include in your picture that give the picture context. For example, if your child is smiling at something besides the camera and you, you may want to include what is being smiled at, unless the smile and face tell the whole story you want to tell. If you're not sure how much to include, remember that you can crop the finished photos later, as long as they are of sufficient resolution and quality, but it's always best to get as close to the exact framing from the start. (Refer to the section Rey-Eye, Cropping, and Photo Cleanup.)