This is the first part of a 3-part series of study on the Chinuch Halachos for the Three Weeks between the 17 Tammuz and 9th of Av.
1. 17 Tammuz is a fast day, and fasting applies to both men and women who are physically capable of fasting. Children below the age of bar/bas mnitzvah are not required to fast on any fast-day, with the exception of Yom Kippur. 
2. There is one restriction which applies to children who have reached the age of understanding on all fast days. Parents should only feed simple foods to these children on these fast days. Delicacies or
specially tasty things should be avoided. In this way, the child is educated to take part in the fasting of the adult community. No such restriction applies to children too young to understand what a fast day is. 
3. Some have a custom, nevertheless, to train children who have reached the age of chinuch to fast part of the day if possible. 
3. Askenazic custom is for adults to refrain from haricutting during the entire three weeks of mourning. Some Acharonim are of the opinion that parents must also refrain from cutting children’s hair during this period. Others disagree and only restrict haircutting for children during the actual week in which Tisha B’Av occurs. Both opinions agree that the restriction of haircutting applies even to children below the age of understanding. 
4. If a child becomes three years old during the Three Weeks, the haircut should be postponed until after Tisha B’Av; alternatively the child’s hair should be cut prior to 17 Tammuz. 
note: Law #4 is from source , and it doesn’t go into as much detail as source . IMHO, it applies to those following the stricter rule, see Law #3. Ask your Rav.
5. Askenazic communities also refrain from attending musical events or programs for the entire three wekk period. Despite this, children may be taken to sports events or programs where music is incidental. This is even permitted during the nine days of mourning immediately prior to Tisha B’Av. Of course, actual concerts or musical performances should be avoided. 
6. Children who have reached the age of chinuch (i.e. 7 or 8 yo) should not listen to music during the entire Three Weeks. 
 A Parent’s Guide to Teaching Children Mitzvot by Rabbi Shmuel Singer
 Children In Halachah by Rabbi Simcha Bunim Cohen.
A resource for Jewish families that contains a chart of the various activities affected by the Three Weeks can be found here .