Why did Moshe instruct Aharon to take specifically a calf as a Chatas?
Rashi: It was to publicize that Hakadosh-Baruch-Hu was pardoning Aharon for making the Golden Calf. 1
Ramban #1 (citing the Toras Kohanim), Moshav Zekenim #1: It was to atone for his participation in the sin of the Golden Calf. 2
Targum Yonasan: It was to prevent the Satan from prosecuting Aharon for his participation in the sin of the Golden Calf. The ram for an Olah was, so that Hashem should recall on his behalf the merit of Yitzchak, whose father bound him like a ram on Har ha'Mori'ah.
Moshav Zekenim #2: Egel Ben Bakar corresponds to Avraham, who took "Ben Bakar" (Bereishis 18:7) for his guests. The ram Olah corresponds to [what Avraham offered in place of] Yitzchak; also, the Gematriya of "Ayil l'Olah" equals that of Yakov. They were Temimim, corresponding to Yakov "Ish Tam."
Rosh, Da'as Zekenim: The Kehunah was taken from [Aharon] through a calf, and through a calf it was returned to him.
Why, in the following Pasuk, were the people told to take specifically a goat as a Chatas and a calf and a lamb as an Olah?
Targum Yonasan (3): The goat, to prevent the Satan - who is compared to a goat - from prosecuting them for the kid-goat that they slaughtered when Yakov's sons sold Yosef in order to trick their father; the calf, because they worshipped the Golden Calf, and the lamb, so that Hashem should recall on their behalf the merit of Yitzchak, who was bound like a lamb.
Moshav Zekenim: Great is the power of a Tzibur. Hashem did not want to shame them to bring a calf for a Chatas like Aharon did (it clearly atones for the Egel 3 ), rather, a goat, like others obligated a Chatas. For idolatry, one must bring a goat (and not a lamb). They were Az (brazen) to say "Eleh Elokecha Yisrael", therefore they bring a Se'ir Izim. However, He commanded to bring as Olos a calf with a lamb. He desired their Teshuvah so much that he chose a lamb, which is brought for the Tamid, to bring with the calf. 4 Or, the lamb corresponds to Yakov and the Shevatim, since Aharon did not bring corresponding to them.
Da'as Zekenim (3): Since they were told to take a calf, they knew that they were pardoned for the Golden Calf.
All the Korbanos listed here are equivalent to the Korbanos of the Kohen Gadol and the people on Yom Kipur - which came to atone for the same sins (Ramban).
Moshav Zekenim: It says "Se'ir Izim l'Chatas v'Egel" to hint that it is for Chet ha'Egel (but it is not overt).
How does this show that He desires their Teshuvah? Perhaps the Tamid shows that the closeness of Yisrael to Hashem; bringing these together shows that Hashem wants to return to the same closeness. (PF)
Where was Moshe commanded to instruct Aharon and the people to bring these Korbanos?
Why were the Korbanos listed here not mentioned in Shemos (29:1) together with the Korbanos that were destined to be brought on the occasion of the Milu'im?
Ramban #1, Moshav Zekenim: Because whereas the Korbanos listed there were in honor of the Shiv'as Yemei ha'Milu'im, those listed here on the eighth day were brought by the Kohanim themselves to inaugurate them. 1
Ramban #2: The Korbanos here were to atone for the sin of the Golden Calf, 2 which took place after the initial command to build the Mishkan was issued.