First of all, Peter being the first bishop of Rome is a side topic of apostolic succession. There were many Orthodox Patriarches from the Ecumenical Patriarch of Constantinople, the Greek Orthodox Patriarch, Antiochian, etc. All of them believe in apostolic succession. And from the statements in regards to the Orthodox Churches as it touches on the subject of Peter being a bishop of Rome, none deny it. The best source in regards to the succession comes from Tertullian's writings where he argues that the gnostics do not have any discipleship connection through the Apostles. You will recall there were many works floating around by gnostics teaching that the material world was evil, that God of the O.T. was full of wrath (Marcion), and many other such heresies. If you want to discuss Peter's bishopric over Rome, I'll be more than happy to cut and paste Tertullian's writings as well as Eusibus' "Ecclesiastical History" in another forum.
I do not feel it necessary to discuss baptism in connection to apostolic succession, unless there is some specific passage you have in mind that might tie it together from the scriptures or from the Early Church Fathers (Ante-Nicene, Nicene, or Post-Nicene Fathers).
I will introduce a scriptural passage that I think touches on the distinction of specifically an apostolic succession from other bishops. The Thomas-Nelson "Orthodox Study Bible" (SAAS-NKJV) has in its footnote four positions in Church government: Bishop, Presbyter, Deacon, and Laity. What is really interesting is the passage in Acts 1.25.
25 to take the placeg in this ministry and apostleship from which Judas turned aside to go to his own place.”
The Holy Bible : New Revised Standard Version, Catholic Edition. 2008 (Ac 1:25). Collegeville, MN: Liturgical Press.
ἀποστολή, ῆς, ἡ apostolē apostolic authority; apostolic office
Balz, H. R., & Schneider, G. (1990-). Vol. 1: Exegetical dictionary of the New Testament (142). Grand Rapids, Mich.: Eerdmans.
1:12–26 Matthias chosen to restore the twelve apostles. The names of the eleven apostles left after Judas’ betrayal come in a different order than in Luke 6:14–16. The Gospel list seems to follow the order in the list as Luke received it. However, Acts adjusts the list in order of importance. It names Peter first, then John (his partner in Acts 3–5), then James, and fourth Andrew. Luke 6:14 had “Peter and Andrew, James and John.”
There were at least three important men called James. The brother of John, one of the Twelve, was with Jesus at the transfiguration and agony in the garden and was killed in Acts 12:2. He is often known as “James the Great.” The son of Alphaeus also belonged to the Twelve. Tradition calls him “James the Less.” And there is the brother or relative of Jesus who had become leader of the Jerusalem church (see Acts 12:17, Acts 15 and 21.)
Little is known about most of the Twelve aside from lists of their names. Most do not appear elsewhere in the New Testament. They are less important as individual personalities than as members of the Twelve. Jesus promised that the twelve tribes of Israel would be restored and that the Twelve would rule them in God’s kingdom (Luke 22:28–29). To have twelve rulers, Judas had to be replaced. Acts 1:15–36 shows that all twelve are in place by Pentecost, when the Holy Spirit empowered them to become the new leaders of God’s people in Jesus’ name.
Bergant, D., & Karris, R. J. (1989). The Collegeville Bible commentary : Based on the New American Bible with revised New Testament (1038). Collegeville, Minn.: Liturgical Press.
Now, you are going to find that the apostles traveled and established churches and left bishops to take charge in the stead of the apostle's authority. This is apostolic succession. When the apostles died, the epicenters of Christian influence: Antioch, Alexanderia, Rome, Jerusalem, and others filled that void through diverse manners of elections fitting their circumstance and custom.
