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The world we live in is full of shortcomings. In working to perfect it, we rush around all week long. However, on Shabbat, a preview of the World to Come, we should look at the good in the world through the lenses of faith and tranquility, and relax. We should imagine everything already perfected, leaving us with no further need to make haste. There is a mitzva to express this spiritual perspective by walking at a leisurely pace on Shabbat. Therefore, running, and even walking fast, are prohibited on Shabbat.
However, if one is going to a Torah class or to prayers, it is a mitzva to walk quickly or even run, as running to do a mitzva does not detract from the honor due to Shabbat. On the contrary, it expresses the spirit of Shabbat, which not only gives us a break from the troubles of the world but energizes and motivates us to advance in our Torah study and mitzva observance.
Running is permitted on Shabbat to escape the rain or to watch something enjoyable. Children and teens who enjoy running may participate in games which involve running. Adults may jump as part of playing with small children. A person may walk on Shabbat for health reasons, on condition one does not walk briskly but at a leisurely, normal pace.
Riding a bicycle on Shabbat is prohibited, as is playing soccer, basketball, tennis, and other sports, because these are considered weekday activities.
