Answer:  There is one main passage that deals with the priesthood of all believers.  It is as follows: "You also, as living stones, are being built up a spiritual house, a holy priesthood, to offer up spiritual sacrifices acceptable to God through Jesus Christ. Therefore it is also contained in the Scripture, "Behold, I lay in Zion A chief cornerstone, elect, precious, And he who believes on Him will by no means be put to shame." Therefore, to you who believe, He is precious; but to those who are disobedient, "The stone which the builders rejected Has become the chief cornerstone," and "A stone of stumbling And a rock of offense." They stumble, being disobedient to the word, to which they also were appointed. But you are a chosen generation, a royal priesthood, a holy nation, His own special people, that you may proclaim the praises of Him who called you out of darkness into His marvelous light;" (1 Peter 2:5-9).

Old Testament priests were chosen by God, not self-appointed; and they were chosen for a purpose: to serve God with their lives by offering up sacrifices first in the tabernacle and then later in the temple.  Later, David also appointed some of them to praise God in song as well.  The Temple Service served as a picture or "type" of the coming ministry of Jesus Christ...a picture that was then no longer needed once His sacrifice on the cross was completed.  When the thick temple veil that covered the doorway to the Holy of Holies where God's presence dwelt above the Mercy Seat (the lid covering the ark of the covenant) was torn in two by God at the time of Christ's death on the cross for our sin (Matthew 27:51), God was indicating that the Old Testament priesthood was no longer necessary...now people could come directly to God through the great High Priest, Jesus Christ, who serves the Father in Heaven (Hebrews 4:14-16).  There are now no earthly mediators between God and man as existed in the Old Testament priesthood.

Christ our High Priest has made one sacrifice for sin for all time (Hebrews 10:12), and there is no more sacrifice for sin that can be made (Hebrews 10:26).  But as priests once offered other kinds of sacrifices in the temple (free will offerings, etc.), so it is clear from 1 Peter 2:5,9 that God has chosen Christians "to offer up spiritual sacrifices acceptable to God through Jesus Christ."  1 Peter 2:5-9 speaks of two aspects of the priesthood of the believer.  The first is that believers are privileged.  To be chosen by God to be a priest was a privilege.  And all believers have been chosen by God: a "chosen generation...His own special people" (verse 9).  In the Old Testament tabernacle and temple, there were places where only the priests could go, and into the Holy of Holies, behind a thick veil, only the High Priest could go and that only once a year on the Day of Atonement when he made a sin offering on behalf of all of the people.  But as mentioned above, because of Jesus' death upon the cross of Calvary, all believers now have direct access to the throne of God through Jesus Christ our great High Priest (Hebrews 4:14-16).  What a privilege to be able to access the very throne of God directly, not through any earthly priest.  When Christ returns and the New Jerusalem comes to earth (Revelation 21), believers will see God face-to-face and will serve Him there (Revelation 22:3-4)...again, what a privilege especially for us who were once "not a people"..."without hope"...destined for destruction because of our sin.

The second aspect of the believer's priesthood is that we are chosen for a purpose: to offer up spiritual sacrifices (see Hebrews 13:15-16 for example) and to proclaim the praises of Him who called us out of darkness into His marvelous light.  Thus by both life (1 Peter 2:5; Titus 2:11-14; Ephesians 2:10) and by word (1 Peter 2:9; 3:15) our purpose is to serve God.  As the believer's body is the temple of the Holy Spirit who indwells every believer (1 Corinthians 6:19-20), so God has called us to serve Him from our heart by first of all offering our lives as living sacrifices, separated from the world's defilements that we might live for the One who gave His life for us (Romans 12:1-2).  And one day we will be serving God in eternity (Revelation 22:3-4), but not in any temple, for "the Lord God Almighty and the Lamb are its temple" (Revelation 21:22) (I believe this statement is a reflection of the closeness with God that we will enjoy).  And as the Old Testament priesthood was to be free of defilement, as symbolized by being ceremonially clean, so as Christ has made us holy positionally before the Father, He calls on us to live holy lives that we might also be a "holy priesthood" (1 Peter 2:5).

In summary, believers are called "Kings and Priests" or a "royal priesthood" as a reflection of their privileged status as heirs to the kingdom and servants of the Almighty God and of the Lamb.  And because of this privileged closeness with God, no other earthly mediator is necessary.  Secondly, believers are called priests because salvation is not merely a fire escape from hell, that once believers have it, they can then do their "own thing."  Rather, believers are called by God to serve Him by offering up spiritual sacrifices, i.e., being a people zealous for good works and by giving praise to the One who has given us the great gifts of His Son's sacrifice on our behalf, His love, forgiveness, adoption into His family, etc., all for us, an undeserving people.

Let us journey on... as priests... for the Kingdom...