Why does the Torah write that Moshe set up the Mishkan on Rosh Chodesh Nisan (the eighth day of the Milu'im), when in fact, he first set it up on the first day of the Milu'im, seven days earlier?
In fact, Hashem had instructed him to set it up (See Shemos 26:30), as soon as it was completed, and he extrapolated from this Pasuk that when he would set it up prior to Rosh Chodesh Nisan, it would not be on a permanent basis, but that he would have to take it down again each morning (some opinions say twice or even three times a day [See Ramban DH 'be'Yom ha'Chodesh' and 'u'Kevar Darshu']).
Because from now on, when they would travel, it would be the Levi'im who dismantled it and set it up (Ramban).
Why was Moshe required to dismantle the Mishkan each morning (Ramban) before putting it up, each day during the seven days of Milu'im?
Ramban: Either to teach the Levi'im how it had to be done, or in order to crown Rosh Chodesh with the crown of the inauguration. 1
See opening Rashi in Shemini. See also Ramban for a number of reasons as to why the Mishkan had to remain intact throughout the day and the following night, and could o nly be dismantled first thing in the morning.