For example, After the kid goes to the school, she has more "free time" or "leisure time" or "spare time". Are these words all correct in this sentence? If so, what is difference and which one is most common in colloquial English?
They're all correct. "Spare time" and "free time" are the most common, but "leisure time" is more specific -- it means that you're not working. Free time and spare time usually is taken to mean "time that I'm not on-duty for my regular job," or "time when I'm not at school" for a youngster. So it would be normal to say "I need to use some of my free time to cut the grass," but usually when someone refers to "leisure time" they're doing something they enjoy during that time, not chores.