Obligation of Guests

The obligation of lighting for a guest and one who is traveling:[1]

A. A married guest whose wife [or husband] has remained home:

If one is married and is a guest in someone’s house during Chanukah without his wife, then if his wife is not lighting the candles at home he is obligated to [light his own Menorah[2]] or join in the expenses of his hosts Menorah lighting. However if his wife is lighting the Menorah in his home, [and the guest does not have his own room[3]] then he is not obligated to light the candles.[4] If however the guest has his own room then he is to light with a blessing.[5] Nevertheless, even in a case that the guest is exempt from lighting, the guest is to participate in the lighting of the owner of the house and hear the blessings from the owner, or alternatively the guest is to light his own candles. It is permitted for the guest to light himself with a blessing even if his wife is lighting at home[6], however in such a case he must have in mind to not be included in his wife’s lighting.[7] Nevertheless it is better for him to hear the blessing from another person rather than say the blessing himself. Alternatively he should light the candles before his wife lights.[8] Practically it is no longer the custom today for guests to rely on the lighting of their wives and rather they light on their own with a blessing as explained above.[9]

Summary:

One who is a guest during Chanukah in another person’s house is to light his own set of candles and not rely on the lighting of his wife. In such a case he is to have in mind not to fulfill his obligation with his wife’s lighting or he is to light prior to the time that his wife lights.

Hotel:

Some Poskim[10] rule that if one is a guest in a hotel then he must light his own candles even if his wife is lighting for him at home. The same applies if the wife is the guest and her husband is lighting for her at home.

Hospital:

Some Poskim[11] rule that if one is a patient in a hospital then he must light his own candles even if his wife is lighting for him at home. The same applies if the wife is the patient, such as women after birth, and her husband is lighting for her at home.

B. An unmarried guest or a couple who traveled together:

A single guest, or a couple which are guests in another’s home during Chanukah, is obligated to light their own candles unless the guest relies on the host for his food on a steady basis.[12]

C. Joining the head of the household in his lighting:

In all cases that a guest is required to light candles, an alternative to the guest lighting his own Menorah is for the guest to give a Perutah worth of money to the host to acquire a portion of the oil which the host will use.[13] Some Poskim[14] rule that in such a case the host must add extra oil in his candle on behalf of his guest. Alternatively the host may give him part of the oil as a present, through doing a form of acquisition with the guest.[15] Whenever the guest is fulfilling his obligation through acquiring some of the oil of the host, he must be present while the host lights and recites the blessings over the Menorah.[16] In any event, if the guest is obligated to light, it is best for the guest to light himself rather than acquire some of the oil of the host.[17] 

D. One who is in an area without Jews:[18]

One who is in an area without Jews is to light the Chanukah candles with a blessing even if he is married and his wife is lighting for him elsewhere. [Nevertheless in such a case he is to have in mind to not fulfill his obligation with his wife’s lighting.[19]]

Summary:

A male guest is to light candles on their own and not fulfill their obligation with the head of the household. If the guest is married he should have in mind not to fulfill his obligation with his wife’s lighting, and he is then able to light. A female guest is to acquire some of the oil used for the lighting. If however she is married or is a long term guest at the home and is also given meals, then she is not required to acquire any of the oil, and rather fulfills her obligation with the lighting of the head of the household.

Must one who is traveling on an airplane or train/bus light Chanukah Candles?[20]

Yes.

If one stayed by a host for Shabbos, where should they light the Menorah after Shabbos?

Some[21] say it is to be lit by the host, unless they will be returning home right after Shabbos.