Why was Avraham traveling constantly southwards?
Seforno: He traveled due south, so that the people from both aforementioned towns would have constant access to him - since he saw that some of them had already been attracted to him.
Kli Yakar: One who want to grow wise should tilt southwards (in Shemoneh Esreh, and the same applies here).
Ha'amek Davar: He saw that the south is better for Korbanos, and all the more so for Torah.
Why does it say "going and traveling"?
Mishnas R. Aharon (Vol. 1, p. 175): One must always go from level to level. It says southwards, for one who wants to grow wise should tilt southwards (in Shemoneh Esreh).
QUESTIONS ON RASHI
Rashi writes: "'Journeying, travelling' (Haloch v'Naso'a) - in segments; [Avraham] would dwell in one place for a month or more, and then travel onwards...." How do we derive that he travelled in this manner?
Gur Aryeh: The verbs Haloch, v'Naso'a, are in the infinitive, indicating a process that is present and ongoing.
Rashi writes: "... for a month or more." How does Rashi know that each stop was for no less than one month?
Gur Aryeh: A person who dwells in a place for a month is considered a local resident for purposes of Nedarim (Bava Basra 8a). Had Avraham's encampments been for less than a month, they would have been considered one long journey rather than many consecutive ones. 1
Rashi writes: "Towards the south - of Eretz Yisrael; this is Yerushalayim, in the territory of Yehudah." Isn't Yerushalayim positioned in the middle of Eretz Yisrael?
Gur Aryeh: Yehudah received the south of Eretz Yisrael, and [most of the] Beis ha'Mikdash was in Yehudah's territory, 1 therefore Yerushalayim is considered to be in the south.
Although the division of the land among the Tribes had not yet occurred, the Torah commonly describes locations by future people or events. (See Rashi to 2:14; 14:7.) (CS)