Why does the Torah refer to the generations of Moshe and Aharon, but list only those of Aharon?
Rashi and Targum Yonasan (on Pasuk 2): Because Moshe taught the sons of Aharon Torah, 1 and Chazal have taught 2 that 'The Torah considers whoever teaches his friend's son Torah, is considered as if he fathered him'. 3
Ramban #1: Because having finished counting Yisrael, Moshe is about to proceed counting the Levi'im, The Torah begins with the leaders of the tribe - Moshe and Aharon.
Ramban #2 and Rashbam: Following the list of the descendents of Yisrael, the Torah is currently concerned with the descendents of Aharon (the Kohanim), following which it will list the descendents of the Levi'im; Moshe is included in "Mishpachas ha'Amrami" (Pasuk 27). 4
Why does the Torah see fit to mention "the day that Hashem spoke to Moshe"?
Rashi: Because that is when Moshe taught Aharon son's and is considered as having fathered them.
Rashbam, Da'as Zekenim, Hadar Zekenim: Because, at that stage, before the Mishkan was erected, 1 Aharon's four sons - whom the Pasuk is about to mention - were all still alive. Ramban - And the Torah mentions it here to point out that Aharon and his sons alone were chosen to serve as Kohanim, whereas the rest of the tribe of Levi were chosen only now.
Seforno: Because it is referring to the day on which the tribe of Levi was chosen, some to carry and some to serve him and to bless by His Name (by singing); and on that day Nadav and Avihu were still alive and were therefore included.
Why does it add the words "b'Har Sinai"?
Rashbam: "B'Har Sinai" always refers to before the Mishkan was erected, afterwards, It is called Bamidbar Sinai.