Although there is some good advice here, it's over-complicated.
With your subjects in the shade and a bright background, the camera meters to expose the subjects properly while over-exposing the background. If you had taken the shots with your subjects in the sun as well, the exposure would be more even (both the subjects and the background would expose properly) but you would have harsh lighting because of the sun.
Contrary to what you might think, the flash would actually help in this situation. Firing the flash would make your subjects brighter, and a closer match to the desired exposure of the background. That's called Fill Flash, and is used for just such an occasion.
Sun and shade mixed shots also confuse the white balance of the camera. Your kids probably look washed out because the camera decided most of the light coming in was 'warm' daylight, not the more cool/blue light that you get in shadows. It compensates for this by adding blues to the 'daylight' to make the yellow appear more 'white', which is fine for everything that is lit up by sunlight, but adding more blues to the shadows will make everything in the shadows seem even more blue or 'washed out'.
Often the best conditions for shooting people outdoors are in cloudy weather (nice and soft, diffuse lighting) or as other posters have mentioned, in the hours surrounding sunrise and sunset.
Best of luck! It's a nice camera, but there is definitely a learning curve over a P&S