Because babies dont move around much they are particularly difficult to photograph. One way of getting around this is to get on their level, i.e. lie down next to them with your camera on the floor. This avoids the standard looking down at the baby shot by putting the viewer in the unusual perspective of being on a level with them, looking into their lovely eyes. If the child has the ability to lift its head the effects can be amazing as you almost find yourself looking UP into their eyes.
When photographing toddlers and young children, again the best thing to do is get on their level or even below their level so you are looking up at them. One trick that helps avoid awkwardness is taking pictures of them doing something they enjoy. Get out the paints or a football, or go to the park and head for the climbing frames, swings or slide, or simply ask them to show you some of their favourite toys. All these activities will animate them and banish self consciousness.
Similarly, taking care with the background can either reinforce the atmosphere of the image, or add another dimension to it. For example a photograph of children playing with a park in the background will be completely different to a photograph of children playing on wasteland, or under a table surrounded by adults legs and feet.
Using continuous shooting mode is good with both small and bigger children who are involved in something very active, or who simply like to move around more. The added bonus of this approach is that you can end up with a series of shots that can be brought together to tell a story.
Alter the angles of your shots by using different focal lengths. If you have a digital SLR you could perhaps invest in a zoom lens and a wide angle one, which will give you more options. A decent zoom lens will get you the close ups while minimising the risk of the children becoming self conscious or distracted by the camera. A wide angle focal length can open up angles and gives you a range of interesting perspectives. It will be harder to achieve great shots using an automatic compact digital, although one with an optical zoom of minimum 3x can give you some great shots.
The main thing to do, however, is simply have fun. If you are playful in your approach, children will respond. What they want to do, given the choice, is have fun. They will gladly try things out for you if you approach them in the right spirit and when this works, it will result in photographs that are playful and show spiritedness.