John Gill Commentary Leviticus 9:7

"And Moses said unto Aaron, Draw near unto the altar, and offer thy sin-offering, and thy burnt-offering, and make atonement for thyself, and for the people; and offer the oblation of the people, and make atonement for them; as Jehovah commanded." — Leviticus 9:7 (ASV)

And Moses said to Aaron
This is only observed to show, that as Aaron did not take upon himself this office of himself, but was called to it, and invested with it, by the appointment of God, nor did he enter upon it except through the call of God by Moses, in the sight of the congregation.

go to the altar, and offer your sin [offering], and your burnt
offering.
the young calf and ram.

and make an atonement for yourself and for the people.
first for himself, and then for the people; for, as Aben Ezra says, a man cannot atone for another until he is pure from all sin. This is a character only to be found in Christ, our great High Priest, and so he is a proper person to atone for and take away the sins of others.

Hence, the priests under the law, with their sacrifices, could never take away sin really, only typically; and this shows the imperfection of the Levitical priesthood, that the priests of that order were obliged to offer first for their own sins. This our high priest, of another order, needed not to do; see (Hebrews 7:27Hebrews 7:28).

and offer the offering of the people, and make atonement for them.
typical of the true and full atonement made by Christ, when he offered himself without spot to God.

as the Lord commanded.
Aaron to do, and as he commanded Christ, his Son and our surety, the antitype of Aaron, (John 10:18) (14:31).