Rosh Hashana is approaching. Let’s learn/review the halachos of Chinuch for this holiday.
A. LISTENING TO THE SHOFAR
1. The obligation of hearing the Shofar is a TORAH requirement only for MEN. Women are exempt, because this positive mitzvah is related to time.  Nevertheless, a woman who fulfills this mitzvah receives the reward for the voluntary fulfillment of a mitzvah in which she’s exempt.  A more detailed treatment of a woman’s shofar obligation can be found in , since there are some poskim who say that women who have been accustomed to hearing the shofar every year is considered to have taken a vow, and if she will be unable to hear it a particular year, she should anull the vow on Erev Rosh Hashanah. I am only focusing on Chinuch in this posting.
2. The duty of chinuch obligates the parents to ascertain that their SONS hear the shofar blowing. Even though the requirement doesn’t extend to girls because adult women are exempt from hearing the shofar, nevertheless, it is appropriate to accustom girls to hearing the shofar since adult women fulfill a voluntary mitzvah by doing this. 
3. The chinuch obligation to hear the shofar applies only to children who have reached the age of understanding. Young children below the age of chinuch are encouraged by poskim to not be brought to the shofar blowing if they will disturb the adults. However, if they are present, they should be with their mother, so as not to disturb the MEN, who are obligated to hear the shofar. 
4. It is sufficient for a child to hear only 30 shofar blasts. A MAN is obligated to 100.  A woman meets her obligation with 30 blasts. 
5. One should impress upon the child the importance of remaining silent while the shofar is being blown and to listen carefully to the shofar blasts. 
B. CARRYING THE SHOFAR
1. Some poskim hold that the shofar may not be carried through a public domain on Yom Tov for the sole purpose of blowing it for children. Other poskim permit this. 
C. BLOWING THE SHOFAR
1. According to ,
a) it is forbidden for an adult who has fulfilled the mitzvah of shofar to blow any extra blasts on Rosh Hashanah. However, if a BOY of chinuch age was unable to hear the shofar, it is permitted for an adult to blow the shofar for him to fulfill chinuch. When doing so, it is preferable that the boy recite the brochos.
b) It is forbidden for an adult who fulfilled his own obligation to blow shofar for the benefit of a girl who reached the age of chinuch, since women are exempt from hearing the shofar, but merely out of custom, a girl’s chinuch requirement is not sufficient grounds for blowing extra blasts.
2. According to , if children (of the age of chinuch) miss the shofar blowing, it is permitted to blow the shofar for them to fulfill the mitzvah of chinuch.[Moderator’s Note: It appears to me that according to , there’s no distinction. Since there appears to be a conflict, this is a question for your Rav.
I asked Rabbi Aaron Tendler of Baltimore and his response to 1a) is as follows: “The prohibition of blowing additional blasts is only if you do so in a manner in which the impression is created that the additional blasts are required to fulfill the Mitzvah. For any other purpose, it is not a problem.” ]
3. A woman may blow the shofar on behalf of other women, but not on behalf of men, and not on behalf of boys of chinuch age. 
4. A child of chinuch age may not blow shofar on behalf of a woman. 
REFERENCES A Parent’s Guide to Teaching Children Mitzvot by Rabbi Shmuel Singer
 Children In Halachah by Rabbi Simcha Bunim Cohen.
 Halichos bas Yisrael Volume 2 by Rabbi Yitzhak Yaakov Fuchs