Jaume Huguet, “The Last Supper”, c. 1470
There is much evidence in the Bible of a Christian priesthood.
Our Protestant friends (particularly evangelicals) sometimes claim that all Christians are priests, and that there is no special class set apart from others in the Church.
The priesthood as we know it today is not a strong motif in the New Testament. But this can be explained in terms of development of doctrine: some things were understood only in very basic or skeletal terms in the early days of Christianity. This is even true of doctrines accepted by all, such as the Holy Trinity or original sin. The canon of the biblical books was slow to formulate (four centuries).
The authority of Jewish priests was still accepted in early Christianity. The Apostle Paul was presenting offerings in the temple around the year 58 (Acts 21:26), acknowledged the authority of the Jewish high priest, described himself as a Pharisee (Acts 23:5-6), and observed Jewish feasts (Acts 20:6).
That said, one can still find much evidence in the Bible of a Christian priesthood. Jesus entrusts to His disciples a remembrance of the central aspect of the liturgy or Mass (consecration of the bread and wine) at the Last Supper (Lk 22:19: “Do this in remembrance of me”).
These same disciples were (like priests) models of a life wholly devoted to God, as a matter of lifelong calling. Jesus was their sole master (Mt 6:24). There was no turning back in their ministry (Lk 9:62), and they were called to a radical commitment involving even leaving possessions and their entire families:
Matthew 19:27 (RSV) Then Peter said in reply, “Lo, we have left everything and followed you. . . .” (cf. 4:22)
Luke 14:26 If any one comes to me and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be my disciple. (cf. Mt 10:37)
The priest-disciple must accept hardships and privations and embrace self-denial:
Matthew 16:24 Then Jesus told his disciples, “If any man would come after me, let him deny himself and take up his cross and follow me.” (cf. Mt 10:38)
Matthew 8:19-20 And a scribe came up and said to him, “Teacher, I will follow you wherever you go.”
 And Jesus said to him, “Foxes have holes, and birds of the air have nests; but the Son of man has nowhere to lay his head.”
Celibacy, for the sake of undistracted devotion to the Lord, is commended, provided that someone is called to it:
Matthew 19:12 For there are eunuchs who have been so from birth, and there are eunuchs who have been made eunuchs by men, and there are eunuchs who have made themselves eunuchs for the sake of the kingdom of heaven.
1 Corinthians 7:32-35 I want you to be free from anxieties. The unmarried man is anxious about the affairs of the Lord, how to please the Lord;  but the married man is anxious about worldly affairs, how to please his wife,  and his interests are divided. And the unmarried woman or girl is anxious about the affairs of the Lord, how to be holy in body and spirit; but the married woman is anxious about worldly affairs, how to please her husband.  I say this for your own benefit, not to lay any restraint upon you, but to promote good order and to secure your undivided devotion to the Lord.
Priests serve the Body of Christ:
1 Corinthians 3:5 What then is Apol'los? What is Paul? Servants through whom you believed, as the Lord assigned to each.
1 Corinthians 9:19 For though I am free from all men, I have made myself a slave to all, that I might win the more.
2 Corinthians 4:5 For what we preach is not ourselves, but Jesus Christ as Lord, with ourselves as your servants for Jesus' sake.
They dispense sacraments:
1 Corinthians 4:1 This is how one should regard us, as servants of Christ and stewards of the mysteries of God.
James 5:14 Is any among you sick? Let him call for the elders of the church, and let them pray over him, anointing him with oil in the name of the Lord;
These include baptism:
Matthew 28:19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit,
Acts 2:38, 41 And Peter said to them, "Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit. . . .  So those who received his word were baptized, and there were added that day about three thousand souls.
A universal priesthood of “offering” (sacrifice) extending to “every place” in New Testament times was prophesied:
Isaiah 66:18, 21 . . . I am coming to gather all nations and tongues; and they shall come and shall see my glory, . . .  And some of them also I will take for priests and for Levites, says the LORD.
Malachi 1:11 . . . my name is great among the nations, and in every place incense is offered to my name, and a pure offering; . . .
Protestants sometimes cite 1 Peter 2:5, 9 to the effect that all Christians are priests. But Peter was citing Exodus 19:6: “you shall be to me a kingdom of priests and a holy nation.” The problem is that the older passage couldn't possibly have meant that there was no priesthood among the ancient Hebrews, since they clearly had a separate class of priests (Leviticus: chapters 4-7, 13-14).
This is even seen in the same chapter, since Exodus 19:21-24 twice contrasts “priests” and “people.” Thus, it makes much more sense to interpret 1 Peter 2:5 as meaning a separate, holy, “chosen” class of priests. But the notion of “spiritual sacrifices” (faith, praise, giving to others) does apply to all Christians (Phil 2:17; Heb 13:15-16).