Spirit of the Law: Chanukah 139:17-18

(Kitzur Shulchan Aruch 139:17) “On erev Shabbos we first light the Chanukah lights and then the Shabbos lights.”
The Arizal explains that the function of the Chanukah candles is to bring the light down to us. The function of the Shabbos candles is to lift us up to the source of the light. This Shabbos we go even higher than usual because we are bringing the illumination of Chanukah down to us before we pick ourselves up with the light of Shabbos. For this reason, Shabbos Chanukah has especial power to serve as a vehicle for the revelation of the Shechinah, and this uplifts the whole of creation in an amazing way. Perhaps this is why Rav Pinchas of Koretz, zt”l, said that on this Shabbos one can feel an even greater than usual illumination of the otherworldly exaltation that we will grasp in the time of Moshiach!

(Kitzur Shulchan Aruch 139:18) “After Shabbos, one first makes havdalah and only then lights the Chanukah candles.” (The Mishnah Berurah cites alternative opinions who hold that one should first make havdalah and only then light Chanukah candles. The Mishnah Berurah concludes that either way is acceptable.)
The Me’or V’shemesh, zt”l, writes that the second way is preferable. Since we remain in an elevated “Shabbos” state until we make havdalah, it is better to first descend from Shabbos and elevate the weekday with havdalah. Then we can bring on an illumination—the light of Chanukah—during a newly elevated work-week.
There are those, like the Chasam Sofer, who hold the opposite. They believed that it is worthwhile to hold onto the sanctity of Shabbos even another tiny bit of time than to hurry to make havdalah so that one can light the candles. They held this way even though the illumination of Chanukah may well be greater if the menorah is lit during an actual weekday.