I remember once hearing that the difference between a professional photographer and a regular photographer is 6 rolls of film. The secret to taking great photos is practice, patience, and having an eye for the creative. As others have suggested, a photography class can help, but not necessary. To take better photos, you must know your camera (know how to use it), learn composition (and be able to apply that on-the-fly), and watch the light (as photography is all about controlling light). Rule of thirds and getting closer to the children can greatly improve the interest in the photo. Make sure they're not a tiny dot in the photo, but that their face fills the frame. Of course, if you want the background, you probably don't want to fill the frame, but at least make them the main subjects and not the background.
For some ideas, take the children outdoors to a park, beach, or garden near sunset as the sun isn't as harsh and make sure the background isn't cluttered. They'll also have the freedom to move around. If you take pictures during midday, you'll get harsh shadows and squinting eyes. If you must take photos during the day, find some shade, like under a large tree or in a patio, to help soften the shadows. Indoor shots are more difficult, but if you want some natural looking shots indoor, have the children by the window. These are just general tips for portrait photography, and as with everything in art, the rules and tips can be broken if you find a better method that works for you.
Have patience and take lots of photos. It's all about practice and learning as you go. Note what doesn't look right in the photo and try again from another angle/view and you'll improve before you know it.