The Rebbe's guidance on Chanukah — its lessons and its practice (IK = Igros Kodesh letter number):
Spread light outward. Chanukah lights are placed at the entrance, facing outward, to illuminate even places of darkness; so too we must spread Torah and Chassidus beyond our own circles and reach those who are distant (IK 038, 039, 040, 041, 8285, 845, 9417, 9419, 9421, 10627). Chanukah and Yud-Tes Kislev share one mission: to bring the inner light of Torah out to everyone (IK 8559, 8561, 8919, 9258, 3099, 3116, 5901, 8142, 8725). Publicize the miracle, with extra energy in outreach (IK 9257, 9420). Chanukah is a time to use every opportunity for spreading Chassidus, to guide youth activities and communal gatherings (IK 4973, 4987, 5976, 3100, 7145, 8169, 7777).
Always increase. Just as we add a candle each night, we must keep increasing in Torah, mitzvos and goodness — regardless of what we achieved before (IK 1968, 3141, 6616, 9303, 10293, 10624, 7135, 8287, 10617). Chanukah teaches ever-higher standards in mitzvos, "mehadrin min hamehadrin" (IK 125, 1308, 1317, 7525). Increase light even when darkness prevails outside, transforming darkness into light (IK 5950, 5959, 5018, 6619), and act with faith even when resources seem lacking (IK 8289, 5954). Increase Torah study, prayer and Chassidus during Chanukah (IK 8113, 9625, 6644). The light of Chanukah should continue all year (IK 4965, 6575, 10635, 9799). On the two meanings of "Zos Chanukah," see IK 8292.
The inner meaning. The unique light of the Chanukah candles (IK 1957); the miracle of the oil and levels of mesirus nefesh (IK 1316); chochmah and the inner spark of Jewish faith (IK 315); the oil, the Kohen Gadol's seal and holiness (IK 173); the candle's components and the light of mitzvos (IK 10033); the menorah's three points (IK 10295); olive oil and light in public and private (IK 10298); practical mitzvos and levels of Divine influence (IK 7516); miracles and Jewish strength in challenging times (IK 10039, 9063).
Purity and self-sacrifice. Chanukah celebrates purity overcoming impurity and calls for mesirus nefesh in preserving holiness, and for strengthening pure Jewish education (IK 3994, 7511, 7517, 10303, 8274, 8723, 8921). Rededicate your home, as the Temple was rededicated, by purifying it of negative influences (IK 10618, 10620). The courage and faith of Jewish women in the Chanukah story inspire women today (IK 3137, 8735, 8916, 9623, 10036, 9064, 9252); lessons for Beis Rivkah (IK 8730).
Practice. Light close to nightfall; on Erev Shabbos, earlier lighting is allowed (IK 3192), and make sure the candles burn long enough to satisfy all opinions (IK 3115). Electric lights raise halachic questions (IK 3293). Give Chanukah gelt generously, including to those who learn Chassidus in depth, since it teaches the use of one's talents (IK 5933, 5039, 10634, 5977, 10640), and give tzedakah generously and personally (IK 9065); Chanukah is a time for gemilus chesed (IK 9249, 10034). On giving to poor youths collecting during Chanukah, see IK 8917. Hold Chanukah farbrengens for students and families (IK 1290, 3910, 3132).
Health and blessing. Chanukah's lesson applied to recovery and health (IK 5022, 5952, 6615, 7130, 9629, 9630), and to livelihood (IK 7140).
See also: Shabbos and holidays – The chassidic holidays; Jewish outreach – Spreading Chassidus; Bitachon – Joy and overcoming sadness.