How do you increase the strength and endurance of a ten year old?

Regular cardiovascular exercise will increase endurance. Lifting heavy weights (but doing a small number of repetitions) will increase strength. Both require a healthy diet for them to be effective.

Weight-lifting can cause injury if it isn't done right -- but you can do pull-ups (where you lift your own body weight) or press-ups are pretty easy and safe to do anywhere. If you can do more than 10 then do it while carrying a backpack with lots of heavy books in it. For best strength increases you should only be about 10 to 20 reps.

Do cardiovascular exercise (like running, playing sports) daily ... do strengthening exercises every other day to allow muscle tissue to recover and build.