How would running be affected by the difference in the atmosphere and gravity of the Earth and the moon?

Because of the lack of atmosphere on the moon, you could only run on the moon in a space suit that allows you to breathe. That is cumbersome. A slight advantage results from the lack of air resistance. The gravity on the moon is only about one sixth that of Earth, which would allow you to make enormous leaps that you would not be able to make on Earth (you would also fall much more slowly, so those leaps would not necessarily enable you to progress faster). We do not really have enough experience with this to say exactly how fast you could run on the moon.