That is today. However, in the time of the Temple, they would take their lulav (that's how the bunch is called for short, by the name of the most prominent species) even on Shabbat. To accomplish it, they would bring it to the synagogue before the Holiday, and in the morning, each would recognize his lulav and take it into his hands. In fact, the people of Jerusalem would never put the lulav down, all the time that it was possible, out of their love for the mitzvah.
After one has waved his lulav, he can put it back into the vase, which is not prohibited because of the law of Shabbat or Holiday. When a child can wave a lulav, he should be taught to do it. In fact, a child is gradually taught to do a mitzvah once he is ready. When he knows how to talk, the father should teach him Torah. What phrase should he teach first? "The Torah that Moses commanded is the heritage of the congregation of Jacob."
Art: Interior of the Portuguese Synagogue in Amsterdam by Emanuel de Witte