Using the flame of the Chanukah candles to light another candle:
Lighting a mundane candle-the Shamash: It is forbidden to light any mundane candle from an already lit Chanukah candle. This prohibition applies even against lighting a candle that will be used to light the other Chanukah candles. [Hence, one may not light the Shamash from a Chanukah candle.] However, some Poskim rule one may light a mundane candle [i.e. the Shamash] for the sake of lighting other Chanukah candles, so long as there is no concern that it will extinguish prior to having a chance to light the Chanukah candles. Practically, we are stringent in this matter.
Lighting a Chanukah candle directly: It is permitted to light one Chanukah candle directly from another Chanukah candle [such as by stretching the wick of one candle to the flame of another candle]. Nevertheless, the custom is to be stringent regarding Chanukah candles, not to light even other Chanukah candles from an already lit Chanukah candle. [The above allowance only applies towards candles that have not yet been lit, however those candles that were lit and later extinguished are forbidden to be lit from other Chanukah candles, even if the candle extinguished within the half hour. Certainly, if the Shamash extinguished, it may not be relit from a Chanukah candle.]
Other Mitzvah candles: Some opinions rule that Shabbos candles, Shul candles, and Chanukah candles are all considered candles used for a Mitzvah, and one may hence light them from each other. Likewise, the candles lit for learning Torah and for the sake of an ill person, are also considered Mitzvah candles.
After a half hour: The above restrictions only apply while the candles are lit for their Mitzvah [within a half hour after nightfall]. However, after the passing of the time of their Mitzvah [i.e. a half hour after nightfall] they are permitted in benefit, and it is certainly permitted to use them to light other candles. [However, according to those Poskim who rule that so long as the candles are lit one is not to make use of them, here too, one is not to light a mundane candle from these candles even after the passing of a half hour. However a Mitzvah candle may be lit.]
The custom is not to light other Chanukah candles from an already lit Chanukah candle. Certainly, one may not light the Shamash or other item from the Chanukah candle.
May one who is lighting his first candle of the Menorah light it from another person’s candles?
Yes. However, this only applies if the Menorah’s are adjacent to each other, as if they are a distance apart, one may not move the candle to light and then replace it by his Menorah. It was already explained, that the Menorahs of two different people, are to initially be placed in separate areas in the house.
 Michaber 674:1; Shabbos 22a
 Michaber ibid; Rosh in name of Rif; Ramban; Rav in Shabbos ibid
The reason: As it appears as a belittlement to the Mitzvah, as not everyone is aware that one plans to light the Chanukah candle with it. [Levush; Taz 674:1]
 M”B 674:3 in name of Peri Chadash and so is implied from the Michaber ibid; Kaf Hachaim 674:7
 2nd opinion in Michaber ibid; Sefer Haterumos; Ran; Shmuel in Shabbos ibid
 Rama ibid; Kaf Hachaim 674:8 that this applies even according to the Sefaradim
 Peri Chadash end of 673; Shulchan Gavoa 674:1; Kaf Hachaim 674:3; However, one may not remove the wick from the candle and then light it. [Peri Chadash ibid] Certainly one may not move the candle towards the unlit candle in order to light it. [Peri Chadash 674; Mamar Mordechai 674:1; Kaf Hachaim ibid]
 Michaber ibid; Shabbos ibid
 Rama 674:1; Hagahos Maimanis; Mordechai
The reason: As the main Mitzvah is fulfilled with a single candle, while the remaining candles are not considered such a Mitzvah, and therefore the extra candles should not be lit from the obligatory candle. [Rama ibid]
 M”A 674; M”B 674:2 and 7; Kaf Hachaim 674:4
 Rashal 85; Bach; Elya Raba 674:1; Chayeh Adam 154:26; M”B 674:7; Kaf Hachaim 674:11
 Michaber 674:2
 Tur in name of Sefer Hateruma
 Michaber ibid
 Rama ibid
 Rama 674:1; Nimukei Yosef
 See “Other opinions” in Halacha A in footnotes!
 Bach; Peri Chadash; Elya Raba 674:1; M”B 674:8; Kaf Hachaim 674:12
 M”B 674:8; Kaf Hachaim 674:12
 P”M 674 A”A 2
 See Halacha 13!
 See Halacha 2C!
 Kaf Hachaim 674:10