Well, it's always good, as St.Peter tells us to be ready to give an answer to every man that asks a reason for the hope that is in you. (1 Pet 3:15). We are reminded that it is a spiritual work of mercy to instruct the ignorant. Pope Pius XII says it is the highest task of the theologian to demonstrate as applicable from reason and revelation the truths of the Catholic Faith.
Fr.Adrian Fortescue is definitely a good source. He wrote an article for the Catholic Encyclopedia on the
Eastern schism here. The article on
the Pope is also excellent for this purpose.
Briefly, let me give my own view.
1. Scripture. The three famous Petrine texts. They are really incontrovertible, I have found. Mat 16:16-19, Jn 21:15-17 and Lk 22:31:32.
2. Early Tradition. From the very earliest of Apostolic times, we see the Fathers testifying to the pre-eminence of the Holy Roman Church. Pope St.Clement's letter to the Corinthians, St.Ignatius' of Antioch address and St.Irenaeus plain statement all bear this out very plainly.
3. Church Councils: These provide a plenary source of proof that the See of Peter in Rome is Rock and Head of the Church. Pope Damasus plainly lays this down in a local Council in 381 A.D. The words of Peter the legate of the Apostolic See in the Ecuмenical Council of Ephesus and Peter Chrysologous bear this out.
Finally, one of my favorite testimonies is from St.Maximus of Constantinople, one of the great Greek Fathers of the Church who was killed for his confession, in the seventh century,
If the Roman see recognizes Pyrrhus to be not only a reprobate but a heretic, it is certainly plain that everyone who anathematizes those who have rejected Pyrrhus, anathematizes the see of Rome that is, he anathematizes the Catholic Church. I need hardly add that he excommunicates himself also, if indeed he be in communion with the Roman see and the Church of God.... It is not right that one who has been condemned and cast out by the Apostolic see of the city of Rome for his wrong opinions should be named with any kind of honour, until he be received by her, having returned to her — nay, to our Lord — by a pious confession and orthodox faith, by which he can receive holiness and the title of holy....
Let him hasten before all things to satisfy the Roman see, for if it is satisfied all will agree in calling him pious and orthodox. For he only speaks in vain who thinks he ought to persuade or entrap persons like myself, and does not satisfy and implore the blessed pope of the most holy Church of the Romans, that is, the Apostolic see, which from the incarnate Son of God Himself, and also by all holy synods, according to the holy canons and definitions, has received universal and supreme dominion, authority and power of binding and loosing over all the holy Churches of God which are in the whole world — for with it the Word who is above the celestial powers binds and looses in heaven also
The extremities of the earth, and all in every part of it who purely and rightly confess the Lord look directly towards the most holy Roman Church and its confession and faith, as it were to a sun of unfailing light, awaiting from it the bright radiance of the sacred dogmas of our Fathers according to what the six inspired and holy councils have purely and piously decreed, declaring most expressly the symbol of faith. For from the coming down of the incarnate Word amongst us, all the Churches in every part of the world have held that greatest Church alone as their base and foundation, seeing that according to the promise of Christ our Saviour, the gates of hell do never prevail against it, that it has the keys of a right confession and faith in Him, that it opens the true and only religion to such as approach with piety, and shuts up and locks every heretical mouth that speaks injustice against the Most High.