Catholic Answers has some online tracts that deal with this topic. Here is a link to one:
catholic.com/library/Authority_of_the_Pope_Part_1.asp
I think Irenaeus of Lyon and Cyprian of Carthage are the most convincing early Christian witnesses to the primacy of the bishop of Rome (the Pope):
“But since it would be too long to enumerate in such a volume as this the succession of all the churches, we shall confound all those who, in whatever manner, whether through self-satisfaction or vainglory, or through blindness and wicked opinion, assemble other than where it is proper, by pointing out here the successions of the bishops of
the greatest and most ancient church known to all, founded and organized at Rome by the two most glorious apostles, Peter and Paul, that church which has the tradition and the faith which comes down to us after having been announced to men by the apostles.
With that church, because of its superior origin, all the churches must agree, that is, all the faithful in the whole world, and it is in her that the faithful everywhere have maintained the apostolic tradition” (Irenaeus, Against Heresies 3:3:2 [A.D. 189], emphasis added).
"The Lord says to Peter: ‘I say to you,’ he says, ‘that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it. And to you I will give the keys of the kingdom of heaven; and whatever things you bind on earth shall be bound also in heaven, and whatever you loose on earth, they shall be loosed also in heaven’ [Matt. 16:18–19]). … On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. Indeed, the others were also what Peter was
, but a primacy is given to Peter, whereby it is made clear that there is but one Church and one chair. So too, all [the apostles] are shepherds, and the flock is shown to be one, fed by all the apostles in single-minded accord. If someone does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church?" (Cyprian of Carthage, The Unity of the Catholic Church 4; 1st edition [A.D. 251], emphasis added).