Apostolic Succession in Early Christianity
The early churches did talk about apostolic succession, but not the way the Roman Catholics do today.
The Roman Catholics argue that it is the passing down of authority. Peter, and not any other apostles, passed authority down to the first bishop of Rome, and not any other bishops, and then down to the succeeding bishops of the Roman church.
This is not what the early church believed.
To the early churches, apostolic succession was a proof of the preservation of truth within the churches.
Peter, for example, taught the truth—the faith once for all delivered to the saints—to Linus and other elders in Rome around A.D. 60. Linus taught it to Anacletus, Anacletus taught it to Clement, and so forth.
As I have already observed, the Church, having received this preaching and this faith, although scattered throughout the whole world, yet, as if occupying but one house, carefully preserves it. (Irenaeus, Against Heresies I:10:2, emphasis mine)
In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same vivifying faith, which has been preserved in the Church from the apostles until now, and handed down in truth. (ibid. III:3:2, emphasis mine)
And this was just as true in every other church started by an apostle:
To these things all the Asiatic Churches testify, as do also those men who have succeeded Polycarp down to the present time … Then again, the church in Ephesus, founded by Paul and having John remaining with them permanently until the times of Trajan [began his reign in A.D. 98], is a true witness of the tradition of the apostles. (Ibid. III:3:4, emphasis mine)
We have fellowship with the apostolic churches because our doctrine is not in any way different from theirs. This is our witness of truth. … Run over to the apostolic churches, in which the very chairs of the apostles are still preeminent in their places, in which their own authentic writings are read. These utter the voice and represent the face of each of them individually. (Tertullian, Prescription Against Heretics 21,36, emphasis mine)
In the late 2nd and early 3rd century, apostolic succession was a great argument.
After all, who really understands the Scriptures and the message of the apostles? Is it not those who are directly descended—spiritually speaking—from those to whom the apostles committed their message and the churches themselves?
However, to argue that apostolic succession has faithfully and accurately preserved apostolic tradition for two thousand years, including throughout the massively corrupt Middle Ages is quite a different issue.
The following four part series covers the early church's view of apostolic succession in depth. I think you'll enjoy it.
This is one of those "talking head" videos, though the scenery's nice. I recommend listening to them like you would the radio, while you wash the dishes or have some other chore to do.
These are, however, interesting and well researched. You'll get a well-documented picture of the leadership and structure of the early churches that you may not be able to get anywhere else on the internet.
If you would prefer to read about this, my quote page on apostolic succession is thorough and has lots of notes.