Did early Christian writers teach apostolic succession or reject it? How do their teachings on this topic compare with contemporary and significant historical understandings of apostolic succession?
To answer the question, no the early church did not reject "apostolic succession". But to understand the answer, it is necessary to define exactly what the question is; what precisely is "apostolic succession"? For the early church, it meant "teach the same as apostles". It did not mean, what it came to be later redefined as, some sort of sacerdotal succession.
CLASSIC APOSTOLIC SUCCESSION
The classic (earliest) definition of apostolic succession was to "teach the same" as apostles. Thus you might find apostolic succession in a church from Judea to Great Britain to India; all teaching the same as the apostles. It was the binding to "the faith once delivered".
- It is within the power of all, therefore, in every Church, who may wish to see the truth, to contemplate clearly the tradition of the apostles manifested throughout the whole world; and we are in a position to reckon up those who were by the apostles instituted bishops in the Churches, and [to demonstrate] the succession of these men to our own times; those who neither taught nor knew of anything like what these [heretics] rave about Irenaeus AH Book III, III, 1 (bold mine)
In other words, Irenaeus is contrasting the false teachers who taught things unknown with and apart from what apostolic churches were teaching, which was handed down in succession. There was no room for innovation or tradition.
[Polycarp] departed this life, having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true. -ibid, 4
Again, the definition of "apostolic succession" is detailed as handing down that truth which was learned from apostles. Those teachings alone are true.
REDEFINED APOSTOLIC SUCCESSION
The Catholic Church (CC) now defines apostolic succession in this way.
1399 The Eastern churches that are not in full communion with the Catholic Church celebrate the Eucharist with great love. "These Churches, although separated from us, yet possess true sacraments, above all - by apostolic succession - the priesthood and the Eucharist, whereby they are still joined to us in closest intimacy." A certain communion in sacris, and so in the Eucharist, "given suitable circumstances and the approval of Church authority, is not merely possible but is encouraged. -Catechism CC-
In other words, even though Catholics and Orthodox differ on major theological doctrines, CC may still define itself and Orthodox as having "apostolic succession" by virtue of a claim to a type of sacerdotal priesthood.
There's no need to delve into which denomination accurately teaches what the apostles taught because the point is made; that is, "apostolic succession" has been redefined from "teach the same as apostles" to having some type of "physical continuity as priests".
- Wherefore it is incumbent to obey the presbyters who are in the Church,—those who, as I have shown, possess the succession from the apostles; those who, together with the succession of the episcopate, have received the certain gift of truth, according to the good pleasure of the Father. But [it is also incumbent] to hold in suspicion others who depart from the primitive succession, and assemble themselves together in any place whatsoever, [looking upon them] either as heretics of perverse minds, or as schismatics puffed up and self-pleasing, or again as hypocrites, acting thus for the sake of lucre and vainglory. For all these have fallen from the truth. Irenaeus Book IV, XXVI, 2
Somewhere along the historic line, the shift was made from adhere to those who teach the same apostolic truth to obey your priest and bishop who can offer a continuous sacrifice for you.
To have "apostolic succession" as it was defined and practiced in the earliest Christian years, one simply must "teach the same" as apostles.
Another answer has already cited Irenaeus (130-202). The passage cited from Against Heresies - III.III.1 - appears as a reference in Pope Paul VI' Encyclical, Dogmatic Constitution on Divine Revelation: Dei Verbum, which itself is quoted in the Vatican Catechism entry on the subject (I.I.II.75-77).
"None of the Heretics Claim Succession from the Apostles. New Churches Still Apostolic, Because Their Faith is that Which the Apostles Taught and Handed Down. The Heretics Challenged to Show Any Apostolic Credentials."
The text reads:
But if there be any (heresies) which are bold enough to plant themselves in the midst of the apostolic age, that they may thereby seem to have been handed down by the apostles, because they existed in the time of the apostles, we can say: Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that [that first bishop of theirs ] bishop shall be able to show for his ordainer and predecessor some one of the apostles or of apostolic men,—a man, moreover, who continued stedfast with the apostles. For this is the manner in which the apostolic churches transmit their registers: as the church of Smyrna, which records that Polycarp was placed therein by John; as also the church of Rome, which makes Clement to have been ordained in like manner by Peter. In exactly the same way the other churches likewise exhibit (their several worthies), whom, as having been appointed to their episcopal places by apostles, they regard as transmitters of the apostolic seed. Let the heretics contrive something of the same kind. For after their blasphemy, what is there that is unlawful for them (to attempt)? But should they even effect the contrivance, they will not advance a step. For their very doctrine, after comparison with that of the apostles, will declare, by its own diversity and contrariety, that it had for its author neither an apostle nor an apostolic man; because, as the apostles would never have taught things which were self-contradictory, so the apostolic men would not have inculcated teaching different from the apostles, unless they who received their instruction from the apostles went and preached in a contrary manner. To this test, therefore will they be submitted for proof by those churches, who, although they derive not their founder from apostles or apostolic men (as being of much later date, for they are in fact being founded daily), yet, since they agree in the same faith, they are accounted as not less apostolic because they are akin in doctrine.Then let all the heresies, when challenged to these two tests by our apostolic church, offer their proof of how they deem themselves to be apostolic. But in truth they neither are so, nor are they able to prove themselves to be what they are not. Nor are they admitted to peaceful relations and communion by such churches as are in any way connected with apostles, inasmuch as they are in no sense themselves apostolic because of their diversity as to the mysteries of the faith.
It should be noted that Tertullian is not universally accepted as a Church Father, but you asked about early Christian writers and not Church Fathers. I don't think any first millennium Church Father would object to Tertullian's assessment here, though, as demonstrated by Irenaeus.
I read SLM's post and thought it was missing something. So, I did a search for "Hands" on newadvent.org to find what the Church Father's said about the Imposition of Hands, which is certainly part of the modern form of passing on the priesthood, administered by Bishops, whom the most denominations who have Bishops consider to be the successors to the Apostles.
And at the time of early dawn our Lord lifted up His hands, and laid them upon the heads of the eleven disciples, and gave to them the gift of the priesthood.
This is from an early church manuscript (4th century) found in Syria called the "Teaching of the Twelve Apostles" not to be confused with the Didache (which is over 200 years older, but doesn't include this part)
The document goes on to tell of all the other places that the Apostles handed on their priesthood to
The city of Rome, and all Italy, and Spain, and Britain, and Gaul, together with all the rest of the countries round about them, received the apostles' ordination to the priesthood from Simon Cephas, who went up from Antioch; and he was ruler and guide there, in the church which he had built there, and in the places round about it.
So, even if you discredit this document as being a bit older than the apostles, it's not so old that it makes it look like stuffy churchmen invented a sacerdotal priesthood in the dark ages. At the very least, we got it from some Middle Eastern bishop who just assumed this was the way things had always been done. By the time a few generations had passed, the apostles had also passed.