Unwritten traditions are handed down in the Church

How do we know the Holy place of Calvary, or the Holy Sepulchre? Does it not rest on a tradition handed down from father to son? It is written that our Lord was crucified on Calvary, and buried in a tomb, which Joseph hewed out of the rock; (Mt. 27.60) but it is unwritten tradition which identifies these spots, and does more things of the same kind. Whence come the three immersions at baptism, praying with face turned towards the east, and the tradition of the mysteries?* Hence St Paul says, “Therefore, brethren, stand fast, and hold the traditions which you have learned either by word, or by our epistle.” (II Thess. 2.15) As, then, so much has been handed down in the Church, and is observed down to the present day, why disparage images?

St. John Damascene
Apology Against Those Who Decry Holy Images
Part I

The Holy Spirit’s authority is equal to the the Father and Son

From all which we learn that the person of the Holy Spirit was of such authority and dignity, that saving baptism was not complete except by the authority of the most excellent Trinity of them all, i.e., by the naming of Father, Son, and Holy Spirit, and by joining to the unbegotten God the Father, and to His only-begotten Son, the name also of the Holy Spirit.  Who, then, is not amazed at the exceeding majesty of the Holy Spirit, when he hears that he who speaks a word against the Son of man may hope for forgiveness; but that he who is guilty of blasphemy against the Holy Spirit has not forgiveness, either in the present world or in that which is to come!

Origen
On First Principles, Book I
Chapter III

Baptism is a new birth into the Church, our Mother

Therefore, blessed ones, whom the grace of God awaits, when you ascend from that most sacred font of your new birth, and spread your hands for the first time in the house of your mother, together with your brethren, ask from the Father, ask from the Lord, that His own specialties of grace and distributions of gifts may be supplied you. “Ask,” saith He, “and ye shall receive.” Well, you have asked, and have received; you have knocked, and it has been opened to you.  Only, I pray that, when you are asking, you be mindful likewise of Tertullian the sinner.

Tertullian
On Baptism
Chapter XX

Tertullian recommends delaying baptism even though it is permissible to baptize infants

And so, according to the circumstances and disposition, and even age, of each individual, the delay of baptism is preferable; principally, however, in the case of little children. For why is it necessary—if (baptism itself) is not so necessary—that the sponsors likewise should be thrust into danger? Who both themselves, by reason of mortality, may fail to fulfil their promises, and may be disappointed by the development of an evil disposition, in those for whom they stood? The Lord does indeed say, “Forbid them not to come unto me.” Let them “come,” then, while they are growing up; let them “come” while they are learning, while they are learning whither to come; let them become Christians when they have become able to know Christ.  Why does the innocent period of life hasten to the “remission of sins?” More caution will be exercised in worldly matters: so that one who is not trusted with earthly substance is trusted with divine! Let them know how to “ask” for salvation, that you may seem (at least) to have given “to him that asketh.”

Tertullian
On Baptism
Chapter XVIII

Anyone can perform a baptism but it’s proper for the priests and deacons to

For concluding our brief subject, it remains to put you in mind also of the due observance of giving and receiving baptism. Of giving it, the chief priest (who is the bishop) has the right: in the next place, the presbyters and deacons, yet not without the bishop’s authority, on account of the honour of the Church, which being preserved, peace is preserved. Beside these, even laymen have the right; for what is equally received can be equally given. Unless bishops, or priests, or deacons, be on the spot, other disciples are called i.e. to the work. The word of the Lord ought not to be hidden by any: in like manner, too, baptism, which is equally God’s property, can be administered by all. But how much more is the rule of reverence and modesty incumbent on laymen—seeing that these powers belong to their superiors—lest they assume to themselves the specific function of the bishop! Emulation of the episcopal office is the mother of schisms.

Tertullian
On Baptism
Chapter XVII

Baptism and Eucharist are like two united baptisms

These two baptisms He sent out from the wound in His pierced side, in order that they who believed in His blood might be bathed with the water; they who had been bathed in the water might likewise drink the blood. This is the baptism which both stands in lieu of the fontal bathing when that has not been received, and restores it when lost.

Tertullian
On Baptism
Chapter XVI

Baptism is an essential part of the New Covenant

Here, then, those miscreants provoke questions. And so they say, “Baptism is not necessary for them to whom faith is sufficient; for withal, Abraham pleased God by a sacrament of no water, but of faith.” But in all cases it is the later things which have a conclusive force, and the subsequent which prevail over the antecedent. Grant that, in days gone by, there was salvation by means of bare faith, before the passion and resurrection of the Lord. But now that faith has been enlarged, and is become a faith which believes in His nativity, passion, and resurrection, there has been an amplification added to the sacrament, viz., the sealing act of baptism; the clothing, in some sense, of the faith which before was bare, and which cannot exist now without its proper law.

Tertullian
On Baptism
Chapter XIII

After Baptism the laying on of hands and invoking the Holy Spirit is performed

In the next place the hand is laid on us, invoking and inviting the Holy Spirit through benediction.

…Then, over our cleansed and blessed bodies willingly descends from the Father that Holiest Spirit. Over the waters of baptism, recognising as it were His primeval seat…