What distinction did the Kohanim draw between gathering the Kahal and preparing them to travel?
Rashi and Seforno: Whereas the former comprised merely a Teki'ah, the latter comprised Teki'ah, Teru'ah, u'Teki'ah. 1
See Sifsei Chachamim. Seforno: Teru'ah, because the Mikdash was traveling; Teki'ah, because The Kahal was gathering to travel together.
Why did they blow a Teki'ah for gathering the Kahal and add a Teru'ah for preparing them to travel
Ramban: 'Since gathering together merits a Teki'ah, which hints at Midas Rachamim, because the straight note hints at Hashem's Right Hand, which is "stretched out to accept those who do Teshuvah" ... and in Pasuk 10, the Torah writes, "And on the day of your rejoicing ... you shall blow a Teki'ah", because whereas war denotes Midas ha'Din, the Yamim-Tovim and the Simchah denote Rachamim'. This also explains why the walls of Yericho fell as the result of a Teru'ah (See Yehoshua, 6:10 & 20).
Why does the Torah mention blowing the trumpets only in connection with Degel Machaneh Yehudah and Degel Machaneh Reuven (in this Pasuk and in the following one)?
Ramban (citing the Sifri): The Torah adds Teru'a Yisk'u le'Mas'eihem" to teach us that they also blew thee trumpets when Degel Machaneh Efrayim and Degel Machaneh Dan traveled.
Seforno: Because the carriers of the Mishkan (Gershon and Merari) traveled together with Machaneh Yehudah, and the carriers of the Mikdash (K'has) 1 , together with Nachaneh Reuven.
How many times did they blow (Teki'ah, Teru'ah, u'Teki'ah) for each of the two remaining camps?
Ramban: Some say only once for both of them; according to others, they blew twice, once for each camp.