Here are some famous texts to prove how Tradition was understood in the first centuries:
Tertulian wrote about 200 A.D. his book The Prescription against Heretics:
“(The Father sent His Son who in turn sent His Apostles:) Immediately, therefore, so did the apostles, whom this designation indicates as the sent. (…) After first bearing witness to the faith in Jesus Christ throughout Judæa, and founding churches (there), they next went forth into the world and preached the same doctrine of the same faith to the nations. They then in like manner founded churches in every city, from which all the other churches, one after another, derived the tradition of the faith, and the seeds of doctrine, and are every day deriving them, that they may become churches. Indeed, it is on this account only that they will be able to deem themselves apostolic, as being the offspring of apostolic churches. Every sort of thing must necessarily revert to its original for its classification. Therefore the churches, although they are so many and so great, comprise but the one primitive church, (founded) by the apostles, from which they all (spring). In this way all are primitive, and all are apostolic, while they are all proved to be one, in (unbroken) unity, by their peaceful communion, and title of brotherhood, and bond of hospitality — privileges which no other rule directs than the one tradition of the selfsame mystery.
“From this, therefore, do we draw up our rule. Since the Lord Jesus Christ sent the apostles to preach, (our rule is) that no others ought to be received as preachers than those whom Christ appointed; for no man knows the Father save the Son, and he to whomever the Son will reveal Him. Matthew 11:27 Nor does the Son seem to have revealed Him to any other than the apostles, whom He sent forth to preach — that, of course, which He revealed to them. Now, what that was which they preached — in other words, what it was which Christ revealed to them — can, as I must here likewise prescribe, properly be proved in no other way than by those very churches which the apostles founded in person, by declaring the gospel to them directly themselves, both vivâ voce, as the phrase is, and subsequently by their epistles. If, then, these things are so, it is in the same degree manifest that all doctrine which agrees with the apostolic churches— those molds and original sources of the faith must be reckoned for truth, as undoubtedly containing that which the (said) churches received from the apostles, the apostles from Christ, Christ from God. Whereas all doctrine must be prejudged as false which savors of contrariety to the truth of the churches and apostles of Christ and God. It remains, then, that we demonstrate whether this doctrine of ours, of which we have now given the rule, has its origin in the tradition of the apostles, and whether all other doctrines do not ipso facto proceed from falsehood. We hold communion with the apostolic churches because our doctrine is in no respect different from theirs. This is our witness of truth.” (Chapters 20 and 21) http://www.newadvent.org/fathers/0311.htm
St Vincent of Lerins, in his Commonitory, (434 A.D.) gives the golden rule of the Catholic faith:
“Moreover, in the Catholic Church itself, all possible care must be taken, that we hold that faith which has been believed everywhere, always, by all (ubique, semper et ab omnibus). For that is truly and in the strictest sense Catholic, which, as the name itself and the reason of the thing declare, comprehends all universally. This rule we shall observe if we follow universality, antiquity, consent. We shall follow universality if we confess that one faith to be true, which the whole Church throughout the world confesses; antiquity, if we in no wise depart from those interpretations which it is manifest were notoriously held by our holy ancestors and fathers; consent, in like manner, if in antiquity itself we adhere to the consentient definitions and determinations of all, or at the least of almost all priests and doctors.” http://www.newadvent.org/fathers/3506.htm
These texts from the early Fathers and Christian writers give the clear teaching of the first centuries of the Church on the nature of Tradition.