Zevachim 115: The Firstborn and the Priests in the Evolving Sacrificial Rites

The bechorim, firstborn, performed the sacrifices on private altars.  Once the Tabernacle was built, sacrifices were brought there the rites were performed by the priests who descended from Aaron.  Our last Mishna (112) taught that sanctification of the firstborn was given at the exodus (Shmot 13:2).  Firstborns redeemed themselves from the Temple service later (Bamidbar 3:12).  The Jerusalem Talmud suggests that the tribe of Levi did not participate in the incident of the Golden Calf, which is why they were given the honour of sacrificing.

A baraita is taught by Rabbi Asi: the firstborn sacrificed for one day.  Like at Sinai, the sacrifices were brought in one, two years and three months after the exodus.  Once the Tabernacle was built (two years and one month after the exodus), Nadav and Avihu were the priests.  They died tragically, and it was only after that time that Aaron's other descendants took over the sacrificial services in the Temple.

Rashi reads this differently: the "young men" described in Shmot (24:5) did not bring the sacrifices themselves.  Instead they represented the people as witnesses to the service. Rashi suggests that the firstborn never did have a role in bringing sacrifices.