Which staff did Hashem instruct Moshe to take?
Rashbam: Aharon's staff that was placed in front of the Aron ha'Kodesh. 1
Targum Yonasan: The staff of miracles (belonging to Moshe). 2
The words "mi'Lifnei Hashem" in Pasuk 9 bear this out. See Ibn Ezra there.
The staff with which Moshe performed all the miracles. See Na'ar Yonasan, who elaborates and who points out that this is also the opinion of the Zohar. Presumably, this is the staff to bring water earlier - the staff with which he struck the river (Sh'mos 17:5). (PF).
Why did Hashem tell Moshe to take the staff?
Seforno, Da'as Zekenim, Hadar Zekenim and Targum Yonasan: In order to strike the rock (in addition to speaking to it 3 ? Seforno and Targum Yonasan).
What did Hashem instruct Moshe and Aharon to do to the rock?
Rashi (on Pasuk 11), Ramban #1 (citing the Midrash), Rashbam and Targum Onkelos: He instructed them to speak to it. 1
Ramban #2: He instructed them to speak to the people about the rock 2 (to inform them that it will produce water), which he did, and then to actually produce water by striking it. 3
Seforno: He instructed Moshe and Aharon to speak to the rock to produce water from itself (a supreme miracle), 4 and to strike it for the water to flow to the tribes 5 (a lesser miracle). 6
Targum Yonasan: He instructed them to make the rock swear by the Name of Hashem that it would produce water, and that, if it refused, Moshe alone was to strike it with the staff.
Da'as Zekenim and Hadar Zekeni (citing the Bechor Shor): He instructed them to speak to B'nei Yisrael in front of the rock. 7
Rosh (citing R. Dan Ashkenazi): Here and in some other places 'Dibur refers to striking - like they struck it earlier (in Refidim).
See Ramban's objections to this explanation.
Ramban: Not to speak to the rock.
Ramban: Like in Egypt (See Shemos 7:15), where Moshe took his staff in order to perform miracles with it). Otherwise, what was the point in taking it?
Seforno: To demonstrate the greatness of Hashem. Refer to 20:8:3:3**.
See Seforno who elaborates.
This is the same episode mentioned in Sh'mos 17:2-7 (refer to 14:22:151:3). There it says explicitly that Hashem told Moshe to strike the rock.
Why does the Torah mention the cattle?