Order of importance or of action
קַח אֶת הַמַּטֶּה וְהַקְהֵל אֶת הָעֵדָה אַתָּה וְאַהֲרֹן אָחִיךָ וְדִבַּרְתֶּם אֶל הַסֶּלַע לְעֵינֵיהֶם וְנָתַן מֵימָיו וְהוֹצֵאתָ לָהֶם מַיִם מִן הַסֶּלַע וְהִשְׁקִיתָ אֶת הָעֵדָה וְאֶת בְּעִירָם
My grandfather focuses on the last part of the instruction -- to give water to the people and to their livestock. He cites the Torah Temima who cites the Magen avraham on what Chazal said in Gittin 62a. It is prohibited to eat before given one's animals to eat. But that only applies to food. With respect to drink, the peson comes first. We find evidence of that halacha in the way Rivka responded to Eliezer's request for water. She offered to him first and then to the camels. The question is: why do they not bring proof from this verse?
My grandfather suggests that the order offered here does not necessarily prove the halacha as the people are mentioned first because they are of primary importance, and the miracle is performed for them.
He offers other examples of directives given in order of importance rather than chronology, as in the case of Hashem telling Moshe to carve the second set of luchos. Devarim 10:1
פְּסָל לְךָ שְׁנֵי לוּחֹת אֲבָנִים כָּרִאשֹׁנִים וַעֲלֵה אֵלַי הָהָרָה וְעָשִׂיתָ לְּךָ אֲרוֹן עֵץ
"Hew for yourself two stone tablets like the first ones and come up to Me onto the mountain, and make for yourself a wooden ark.
Rashi says: ואמר לי פסל לך, ואחר כך ועשית ארון, ואני עשיתי ארון תחלה, שכשאבוא והלוחות בידי היכן אתנם
He said to me, “Hew for yourself [two tablets],” and afterwards, “make for yourself a [wooden] ark.” I, however (see verse 3), made the ark first (Tanchuma 10), because [I considered that] when I would come with the tablets in my hand, where would I put them?
In the comman, the object of primary importance comes first even if another step has to be done before it. That is why the proof of the halach comes from Rivka who was concerned only with the practical nature of the actions.
Related post: http://kallahmagazine.blogspot.com/2015/10/dvar-yehudah-parsha-points-from-my.html