This depends on the position he plays. If he is a middle infielder (Shortstop or Second Base), you would want to get him a smaller glove to improve transfer of the ball from the glove to his throwing hand - about 9-10 in. If he plays outfield, the his glove should be much bigger to improve catching ability. This is about 11-12 inches. For a first baseman, buy a first baseman's glove (it will say on the tag) that is about 12 inches or longer. Third baseman's gloves should be about 11 inches. A catchers mitt is basically any catcher's mit you can hold up. Pitcher's should be around 10-11 inches with a thick web (pocket). If your player has never played before, he will probably be playing second base or outfield, and for him a large glove is recommended. For the average 13-year-old, a 9-12 inch glove is proper.
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You should get a -11 bat (which means that the weight is 11 ounces less than the amount of inches on the bat) and a glove that's about 10-11 inches.
It depends on the size of your child. Most 7 yr olds are going to wear an 11"-11 1/2" glove. Make sure that you get a glove that is easy for them to squeeze when it is on their hand.
The size of the glove depends on what position she plays. This buyers guide should help:
http://www.hitrunscore.com/softball-glove-buyers-guide.html
It depends on the length of the fingers and sometimes the width but i got an 12". The one that i got works really well. Its a mizuno.