Halacha says that the lighting of the Menorah precedes the lighting of Nerot Shabbat. So on Friday night, we first light the Menorah and then the ladies light the Shabbat candles.
The normal time for Shabbat candle lighting is eighteen minutes before sunset. Menorah lighting on Friday night would then be about 20-23 minutes before sunset. In Brooklyn, New York, Shabbat candle lighting time is at 4:10 PM. Therefore, Menorah lighting time on Friday night is at about 4:05 PM.
Halacha says that one must be careful not to light the Menorah on any given day (including Friday) before a time that is called Plag Mincha. In Brooklyn, New York, Plag Mincha during Chanukah is at about 4:00 PM. So on Friday night, one must light after 4:00 PM and before Shabbat candle lighting time which is at 4:10 PM. If one lit before 4:00 PM (Plag Mincha), he would have to put it out and relight again.
Now we also should mention that one should put enough oil into his Menorah, or have long enough Menorah candles, to last until half hour after sunset. On week nights, the Menorah preferably is lit about 12 minutes after sunset, and it is supposed to last at least 30 minutes. That means the light should normally last at least past a total of 42 minutes after sunset. On Friday night we said above that we light the Menorah about 23 minutes before sunset. So when you add 42 minutes plus 23 minutes, it means that on Friday night the Menorah needs to last for at least about 65 minutes. (Torat HaMoadim page 92)
On Motzae Shabbat (Saturday night) the custom in the synagogue is to light the Menorah first and then make Havdalah. The custom at home however, is to make Havdalah first and then light the Menorah. Why the disparity? We light the Menorah in the synagogue before making Havdalah for primarily 2 reasons. First, we want to be Pirsum HaNes and glorify the miracle of Chanukah with the lighting of the Menorah in the synagogue. But since many congregants leave immediately after Havdalah and would miss Menorah lighting, we therefore light the Menorah first. The second reason is in order to prolong the Shabbat. So for these 2 reasons we light the Menorah before making Havdalah when in shul. But when at home, these reasons do not apply and therefore in the house one would light the Menorah only after making Havdalah. (Torat HaMoadim page 202-209.)
