Prior to Sinai
Before the Mosaic law was ever given, the Bible makes it clear that there were priests whom God appointed to provide spiritual leadership for believers.
Then the king of Salem brought forth bread and wine; and he was the priest of the most high God. And he blessed him, and said, Blessed be Abram of the most high God, possessor of heaven and earth; And blessed be the most high God, which hath delivered thine enemies into thy hand. And he gave him tithes of all.
When Melchizedek is described as being "without father, without mother, without genealogy, having neither beginning of days nor end of life, "[Hebrews 7:3] it is not suggested that he was a biological anomaly, or an angel in human guise. Historically Melchizedek appears to have belonged to a dynasty of priest-kings in which he had both predecessors and successors. (F.F. Bruce, The Epistle to the Hebrews, Eerdmans Publishing, 1979, p. 137)
When Jethro, the priest of Midian, Moses' father-in-law, heard of all that God had done for Moses, and for Israel his people, and that the Lord had brought Israel out of Egypt .... And Jethro rejoiced for all the goodness which the Lord had done to Israel .... And Jethro said, Blessed be the Lord .... And Jethro, Moses' father-in-law, took a burnt offering and sacrifices for God: and Aaron came, and all the elders of Israel, to eat bread with Moses' father-in-law before God.
Exodus 18:1,9,10 & 12
And the Lord said unto him, Away, get thee down, and thou shalt come up, thou, and Aaron with thee; but let not the priests and the people break forth upon them [this occurred before priests of Aaron were called].
Under the Mosaic Law
And take thou unto thee Aaron thy brother, and his sons with him, from among the children of Israel, that he may minister unto me in the priest's office, even Aaron, Nadab and Abihu, Eleazar and Ithamar, Aaron's sons.
And thou shalt anoint them, as thou didst anoint their father, that they may minister unto me in the priest's office: for their anointing shall surely be an everlasting priesthood throughout their generations.
After the cross
In Romans 15, Paul referred to his priestly duty as a minister of Christ:
I have written you quite boldly on some points, as if to remind you of them again, because of the grace God gave me to be a minister of Christ Jesus to the Gentiles with the priestly duty of proclaiming the gospel of God, so that the Gentiles might become an offering acceptable to God, sanctified by the Holy Spirit.
Romans 15:15-16 (NIV)
In the New Testament, even though certain priestly duties continued, Christ established different ministerial functions and offices from those that existed under the Mosaic law.
And he gave some, apostles; and some, prophets; and some, evangelists; and some, pastors and teachers; For the perfecting of the saints, for the work of the ministry, for the edifying of the body of Christ: Till we all come in the unity of the faith, and of the knowledge of the Son of God, unto a perfect man, unto the measure of the stature of the fulness of Christ.
Apostles, prophets, evangelists, pastors and teachers are to remain in Christ's church until the entire church arrives at the measure of the stature of the fullness of Christ. No church has yet done that. So all of these offices are still needed in Christ's church.