Have small kids: I need a camera that turns on quickly, is small enough to carry everywhere and takes great quality pics. Any recommendations?

Have small kids: I need a digital camera that turns on quickly (is there a technical term for this?), is small enough to carry everywhere (do I need to worry about image stabilzation with compact cameras?) and still takes great quality pics (how many megapixesl is enough?) Any recommendations? What is the best camera for me?

Unfortunately, the picture quality of the Stylus 750 is abysmal! I own the Canon Rebel DSLR and the Canon Pwershot A95. My husband bought me this Stylus because my A95 is on its last legs after taking THOUSANDS of pictures. I too have children, and I also teach. I keep a point and shoot handy for all those "kodak moments." The Sylus 750, while being from Olympus (a trusted name in my family), and being water resistant and having image stabilization, takes horrid pictures. My children's hair looked like fake doll's hair, and a poinsettia I photographed under varying light conditions, and with and without flash, NEVER looked anything better than fuzzy. I was appalled at the image quality, and suprised that Olympus would allow this camera to market. There is no viewfinder, and the LCD, though large, is not very good, either. Please avoid this camera!