As we prepare for the holiday of Chanukah, let us review/learn the halachos pertaining to this holiday from a Chinuch standpoint.
A. LIGHTING THE MENORAH
1. It is a positive Rabbinic mitzvah to kindle a light each night of Chanukah, and is enhanced by adding one more light each successivenight for 8 nights.
2. This mitzvah is incumbent upon men and women alike.
3. Ashkenazic practice is that although each member of the family kindles his own menorah, married women fulfill the mitzvah through their husbands and single girls fulfill the mitzvah through their fathers. It is preferable that the female members of the household be present at the kindling and listen to the recitation of the blessings.   [ Moderator’s note: Here is a footnote from reference that explains the Ashkenazic custom more fully…
The Ashkenazic custom of a wife not lighting is in accordancewith the concept ‘ishto k’gufo’, one’s wife is as one’s self. The Chasam Sofer on Tractate Shabbos 21b explains that althougha daughter and her father are not considered one, it is not customary for her to light. There is a tradition from our forefathers, from the days when the Menorah is kindledoutdoors, that single girls do not light. They refrained because it was unseemly (I suppose for modesty reasons) for them to go out in public and stand among the men at the entrances to homes. They stood a distance to hear the blessings. This custom is also followed indoors and single girls do not light. Mishmeres Shalom explains that since themothers fulfill the mitzvah through their husbands, single girls forgo the extra beautification of the mitzvah to avoid slighting a parent since it might appear disrespectful for them to light when their mothers do not. According to this view, agirl whose mother is a widow or a divorcee and lights her own menorah should also light. ]
4. Parents are obligated to train their children in the performance of this mitzvah. It is customary for boys of chinuch age (5 or 6 yo,depending on the child’s intelligence) to
a) kindle their own menorahs, and
b) their menorahs are capable of burning for the full duration of
time required by halacha. 
Halacha requires that the menorah must burn for at least half an hour after nightfall (when 3 stars appear). If one does not have enough oil to burn that long, one should light without a blessing. 
On Erev Shabbos, one should have enough oil or candles long enough to burn until a half-hour after nightfall. 
B. CHILDREN EXEMPTING ADULTS
1. Adults should not rely on the candle lighting of a child of chinuch age to fulfill their obligation, except in a case of necessity. 2. At least one adult in each household should kindle the menorah.
C. LIGHTING IN SHUL
1. The kindling of the menorah in shul does not exempt anyone from their obligation to kindle at home, even to the person who lights the Menorah in the shul. 
2. The shul’s Chanukah menorah may be kindled by a child of chinuch age. 
3. However, if there are individuals who reside in the shul and wish to fulfill their obligation with the shul lights, a child should not kindle the lights. 
D. ADDITIONAL LAWS OF CHANUKAH FOR ADULTS
 A Parent’s Guide to Teaching Children Mitzvot by Rabbi Shmuel Singer
 Kitzur Shulchan Aruch, vol 2 by Rabbi Avrohom Davis