Children under 6 always travel free of charge on hvv services. For children older than 6, there are reduced-price single/day tickets or school passes. These can already be used if children have not yet started school.
The hvv offers special tickets, for instance the hvv Deutschlandticket, for all school students in regular primary or secondary education at a public school. The type of ticket offered, the price and the rules on its use depend on where you live.