I have grandchildren ranging from 3 to 20 years of age. When my youngest two came along (ages 3 and just-turned-6), we needed a good quality digital camera that took good pictures fast - for the very reasons you mentioned.
I have been an amateur photographer most of my life, using primarily manually controlled 35mm cameras with filters; flash; macro, and zoom lenses; etc.; so I have a good handle on quality camera equipment. I have always trusted Kodak for point and shoot cameras. Their quality has always been good, and their image processing has always been top notch, hands down. With Kodak's reputation for quality coupled with affordability, I decided to try Kodak's EasyShare C653 digital camera. It is compact enough to put in a diaper bag or shirt pocket; has a good LCD view screen on the back for quick shot framing; a good zoom lens; and for quick toddler shots with a minimum of blurred images (quick shots often are blurred due to movement of both the subject and the photographer's hands), there is an anti-blur setting on the selector ring around the shutter button. It also has many other neat features, including close-up and video capabilities. It has some memory built into the camera itself, but in the battery compartment, there is also a slot for a memory card. Unlike some zoom cameras with video capability, the lens can be zoomed in or out while in video mode.
As for the picture quality, I have taken very close shots of tiny insects and flowers, as well as indoor portrait shots both with and without a flash in normal lighting, and I have gotten excellent results in all situations. The only shots I had to fiddle with to get right were nighttime landscape shots with the full moon. It doesn't do nearly as well as a film type camera for outdoor night shots, but it did well enough for wallpaper on my computer screen.
So, from an amateur photographer with toddler grandchildren, I recommend the Kodak EasyShare C653 camera, or their current equivalent. I've had mine for a couple of years now, so Kodak may have a new model on the market now.
I got mine for about $125.00 plus tax, and it has been money well spent.
Oh. You might want to consider getting about eight rechargeable batteries and a charger capable of charging some or all of them-or two chargers capable of charging four batteries each to reduce battery costs, especially if you will be using video mode or flash a lot.
I think you'll be well satisfied with Kodak's EasyShare camera. You can also get an EasyShare docking station to dock the camera to your computer or printer, but I have been using the USB cord that came with the camera and the EasyShare software on my computer to download the images from the camera to my hard drive. You don't need the docking station to print the photos or send them by e-mail, or to upload them to the EasyShare or other image hosting service if you choose to go that route.
I hope this helps.