Note: The Shabbos Torah Reading is divided into 7 sections. Each section
is called an Aliya [literally: Go up] since for each Aliya, one person
"goes up" to make a bracha [blessing] on the Torah Reading.
1st & 2nd Aliyot:
Moshe counts all males over the age of 20, and registers them according to their paternal ancestry. The total, not including Shayvet Layvie, was 603,550. The total number of Jews was around 3 million.
The tribes are each assigned their position within the camp, and in the order of the traveling.
In preparation for separating the tribe of Layvie, the Torah establishes Aharon's genealogy. Although the Kohanim were also from the Tribe of Layvie, they were counted by themselves.
Moshe counts the Leviyim according to their three basic families: Gershon, Kehas, and Mirarri. Their individual responsibilities in transporting the Mishkan is stated, and their total was 22,000.
Being that the Leviyim were to be in direct exchange for the first born, Moshe counts all the first born in the rest of the nation. Their total was 22,273. The extra 273 first born are instructed to redeem themselves from Aharon the Kohain for 5 silver shekels.
The Parsha concludes with detailed instructions for the family of Kahas. Their primary responsibility was to transport the vessels of the Mishkan. Only Aharon and his sons were allowed to cover the vessels in preparation for transport. Once covered, the family of Kahas did the actual carrying.