If you're really very worried about someone being injured when you mow your lawn, you could always borrow a couple of sheep to nibble your grass. However, this isn't a very practical solution. An old-fashioned push reel mower, which depends on you for its power, can be a good solution if your lawn is quite small. As long as you keep the blades properly sharpened and keep an eye out for hidden objects in the grass, you can't really push it fast enough for flying stones and the like to be a real problem.
For bigger lawns, most people these days use powered rotary mowers and these can be quite dangerous in the wrong hands or if there are small children running about. With a powered lawn mower, you have to pay more attention to lawn mower safety. Make sure you use the right type of gas for your mower if you are using a gas-powered machine; if it is electrically powered, pay attention to where the cord is so you don't mow over the cord by accident and cut it.
Rotary mowers have throttles with automatic shut offs to prevent injury, but you still shouldn't touch the rotary blade until it has completely stopped spinning; otherwise you could get a nasty injury. For very big lawns, riding mowers are commonly used, but using one is like driving a small vehicle so they're not suitable for use by anyone under 16. It's highly recommended that you wear full-length pants and shoes when mowing, and even steel-toed boots are a good idea if you have a lot of lawn. Carefully clearing your lawn of debris before you mow will help prevent accidents; wearing safety goggles can protect your eyes. Finally, always make sure to clean and maintain your machine properly.