Summary of the laws of Chanuka

 (All times mentioned are for the year 2017 in Jerusalem.)

The earliest time to light the Menora is at Plug-HaMincha - which is at 15:33 - but the Menora has to burn until 30 minutes after dark.

The best time is at sunset (16:36) or nightfall (17:03) depending on family custom.

You can light any time after this - but you cannot make the Brachot then, unless somebody else is awake in the house with you, or you live on a busy street and people are still walking outside and they can see your Chanuka Menora.

The latest time to light would be 4:24 the next morning - 30 minutes before dawn.

Once it's time to light the Menora, one should not do anything else except get organized to light. Even learning Torah and eating is not allowed; though one can have a light snack (fruit or similar).

The only exception is: one can first Daven Maariv - תפילת ערבית - if one lights at nightfall.

After candle lighting one can do anything. However, there is an ancient custom that women do not do any work for the first 30 minutes, to commemorate their pivotal role in the Chanuka victory. ("Work" being anything they cannot do on Yom Tov.)
The custom is to spend these 30 minutes singing Maoz Tzur and other Chanuka songs, relating the miraculous story of Chanuka and discussing the Halachot of Chanuka.

On Friday afternoon one lights Shabbat candles at the usual time (15:58) and one has to light the Menora BEFORE lighting Shabbat candles. (So the Menora has to be lit between 15:34 and 15:58)

The Menora has to burn until 30 minutes after dark - so you'll need oil or candles that will last for at least 70 minutes.

Since you cannot move the Menora on Shabbat, you have to place it in a safe place - but (if possible) it should not be near your Shabbat candles.

On Motzai Shabbat you must wait until Shabbat is over (17:17) and then you make Havdala and light the Menora. Which to do first depends on family custom.