Children may not play soccer, football, baseball, or basketball on Shabbat. Since adults make a big deal out of these games and they have intricate rules and procedures, they are prohibited, as they are considered a prohibited weekday activity. Besides, sometimes these games involve additional prohibited activities, such as preparing the field for a game, driving to the field, registering for the game, and buying and selling tickets or team merchandise. It is also prohibited to watch a soccer or basketball game on Shabbat, because this is a weekday activity. It is even prohibited to play with the balls associated with these sports at home or in a yard, because they are muktzeh and because it is a weekday activity. For the same reasons, all of the above applies to tennis as well.
Children may play and run around for their enjoyment but may not participate in exercise classes (see above 22:8).
Children may play with balls designed for young children, on condition that they play indoors or in a paved yard. However, they may not play on grass or on a dirt yard, out of concern that they will level the ground. They may play table tennis for fun, since that is generally played indoors. There is no need to worry that by allowing children to play with balls at a young age, they will get used to it and will continue doing so as adults, since the permission is limited to balls designed for children, which in any case adults do not play with.
One may not recover a ball from a tree using one’s hand or a stick, as it might lead to breaking a branch. If the ball fell out on its own, one may play with it (19:7 above; SSK 16:8).