Bolded in Irenaeus quote from your post is Book III chapter 3.1. Now what is this tradition he speaks of? He explains in the same chapter, specifically Book III chapter 3.3:
3. The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes. Nor was he alone [in this], for there were many still remaining who had received instructions from the apostles. In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the apostles, proclaiming the one God, omnipotent, the Maker of heaven and earth, the Creator of man, who brought on the deluge, and called Abraham, who led the people from the land of Egypt, spoke with Moses, set forth the law, sent the prophets, and who has prepared fire for the devil and his angels. From this document, whosoever chooses to do so, may learn that He, the Father of our Lord Jesus Christ, was preached by the Churches, and may also understand the apostolic tradition of the Church, since this Epistle is of older date than these men who are now propagating falsehood, and who conjure into existence another god beyond the Creator and the Maker of all existing things.
CHURCH FATHERS: Against Heresies, III.3 (St. Irenaeus)
Matches what we can find in the written Holy Scriptures. His use of 'tradition' here is the rule of faith. Not all of it as he is specifically refuting a Gnostic point by establishing the tradition is we have One God.
Irenaeus does go into more detail on the apostolic tradition:
1. The Church, though dispersed throughout the whole world, even to the ends of the earth, has received from the apostles and their disciples this faith: [She believes] in one God, the Father Almighty, Maker of heaven, and earth, and the sea, and all things that are in them; and in one Christ Jesus, the Son of God, who became incarnate for our salvation; and in the Holy Spirit, who proclaimed through the prophets the dispensations of God, and the advents, and the birth from a virgin, and the passion, and the resurrection from the dead, and the ascension into heaven in the flesh of the beloved Christ Jesus, our Lord, and His [future] manifestation from heaven in the glory of the Father to gather all things in one, Ephesians 1:10 and to raise up anew all flesh of the whole human race, in order that to Christ Jesus, our Lord, and God, and Saviour, and King, according to the will of the invisible Father, every knee should bow, of things in heaven, and things in earth, and things under the earth, and that every tongue should confess Philippians 2:10-11 to Him, and that He should execute just judgment towards all; that He may send spiritual wickednesses, Ephesians 6:12 and the angels who transgressed and became apostates, together with the ungodly, and unrighteous, and wicked, and profane among men, into everlasting fire; but may, in the exercise of His grace, confer immortality on the righteous, and holy, and those who have kept His commandments, and have persevered in His love, some from the beginning [of their Christian course], and others from [the date of] their repentance, and may surround them with everlasting glory.
CHURCH FATHERS: Against Heresies, I.10 (St. Irenaeus)
