"Speak to the entire community of Israel saying: On the tenth of this month each man shall take for himself a lamb." (Ex. 12:3)

G‑d instructed the people to begin preparations for the Passover sacrifice specifically on the tenth of the month. Since they, to varying extents, had been steeped in Egyptian culture, they had to first purify their inner Divine selves of this negativity before undertaking to purify their animal natures.

...this purification process began as G‑d gave the people their first commandment...

Inasmuch as the commandments purify the person performing them, this purification process began as G‑d gave the people their first commandment, to declare a new month when the moon begins its cycle anew. This occurred on the first day of Nissan, and on each of the ensuing nine days, another of the ten faculties of the people's souls (i.e., the three faculties of the intellect and the seven emotions) was refined as another of the ten sefirot illumined the corresponding soul-faculty.

By the tenth day, all ten faculties of Divine soul had been cleansed by the Divine energy emanating from this commandment, and the people were then ready to begin the task of purifying their human/animal souls. G‑d therefore instructed them to take a lamb, a physical correlate of the animal soul, on this day, and begin to prepare it as the Passover sacrifice.