Megillah, Chapter Two, Mishnah Three
The first section of this mishnah deals with a person who travels from a walled city which reads on the 15th of Adar to a town which reads on the 14th or vice versa.
The second section discusses how much of the Megillah must be read on Purim.
1) A resident of a town who has gone to a walled city or a resident of a walled city who has gone to a town, if he is to return to his own place he reads according to the rule of his own place, and if not reads with them.
2) From where does a man read the Megillah and thereby fulfill his obligation?
a) Rabbi Meir says: all of it.
b) Rabbi Judah says: from There was a Jew (Esther 2:5).
c) Rabbi Yose says: from After these things (3:1).
Section one: The mishnah states simply that if a person travels from one type of town to another he retains the custom of the town of his origin if his intention is not to move to his new town. If his intention is not to return to his previous town, then he reads with the new place.
In the Talmud they explain that if he is to return to his own place means if he is to return there that very night, and get there before the morning. If he goes to another town but returns to his own town in the morning, then he celebrates Purim with his own town. But if he goes to another town and is there in the morning, he must celebrate Purim and hear the Megillah on that day.
By the way, as someone who lives in Modiin, which reads on the 14th, and goes to Jerusalem which reads on the 15th, I encounter this issue pretty much every year.
Section two: Today we read the entire book of Esther, but whether this is necessary is debated by the sages. Rabbi Meir says that one has to read the whole thing. Rabbi Judah says that he only has to read from 2:5, where Mordecai is first mentioned. Rabbi Yose says he only has to read from 3:1, where the actual plot by Haman (make a lot of noise when you say this) begins.