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…

 

The newly baptised are immediately annointed with holy oil and freed from sin

After this, when we have issued from the font, we are thoroughly anointed with a blessed unction,—(a practice derived) from the old discipline, wherein on entering the priesthood, men were wont to be anointed with oil from a horn, ever since Aaron was anointed by Moses. Whence Aaron is called “Christ,” from the “chrism,” which is “the unction;” which, when made spiritual, furnished an appropriate name to the Lord, because He was “anointed” with the Spirit by God the Father; as written in the Acts: “For truly they were gathered together in this city against Thy Holy Son whom Thou hast anointed.” Thus, too, in our case, the unction runs carnally, (i.e. on the body,) but profits spiritually; in the same way as the act of baptism itself too is carnal, in that we are plunged in water, but the effect spiritual, in that we are freed from sins.

Tertullian
On Baptism
Chapter VII

We are cleansed through Baptism to receive the Holy Spirit which connects us to the Church

Not that in the waters we obtain the Holy Spirit; but in the water, under (the witness of) the angel, we are cleansed, and prepared for the Holy Spirit. In this case also a type has preceded; for thus was John beforehand the Lord’s forerunner, “preparing His ways.” Thus, too, does the angel, the witness of baptism, “make the paths straight” for the Holy Spirit, who is about to come upon us, by the washing away of sins, which faith, sealed in (the name of) the Father, and the Son, and the Holy Spirit, obtains.

Moreover, after the pledging both of the attestation of faith and the promise of salvation under “three witnesses,” there is added, of necessity, mention of the Church; inasmuch as, wherever there are three, (that is, the Father, the Son, and the Holy Spirit, ) there is the Church, which is a body of three.

Tertullian
On Baptism
Chapter VI

Any kind of natural water is suitable for Baptism when the water is blessed

And accordingly it makes no difference whether a man be washed in a sea or a pool, a stream or a fount, a lake or a trough; nor is there any distinction between those whom John baptized in the Jordan and those whom Peter baptized in the Tiber, unless withal the eunuch whom Philip baptized in the midst of his journeys with chance water, derived (therefrom) more or less of salvation than others. All waters, therefore, in virtue of the pristine privilege of their origin, do, after invocation of God, attain the sacramental power of sanctification; for the Spirit immediately supervenes from the heavens, and rests over the waters, sanctifying them from Himself; and being thus sanctified, they imbibe at the same time the power of sanctifying.

Tertullian
On Baptism
Chapter IV

God gave life to all things through water and through Baptism gives new life

After the world had been hereupon set in order through its elements, when inhabitants were given it, “the waters” were the first to receive the precept “to bring forth living creatures.” Water was the first to produce that which had life, that it might be no wonder in baptism if waters know how to give life.

… it is not to be doubted that God has made the material substance which He has disposed throughout all His products and works, obey Him also in His own peculiar sacraments; that the material substance which governs terrestrial life acts as agent likewise in the celestial.

Tertullian
On Baptism
Chapter III

Baptism is such a simple act but obtains eternal life

There is absolutely nothing which makes men’s minds more obdurate than the simplicity of the divine works which are visible in the act, when compared with the grandeur which is promised thereto in the effect; so that from the very fact, that with so great simplicity, without pomp, without any considerable novelty of preparation, finally, without expense, a man is dipped in water, and amid the utterance of some few words, is sprinkled, and then rises again, not much (or not at all) the cleaner, the consequent attainment of eternity is esteemed the more incredible.

Tertullian
On Baptism
Chapter II

Christians are reborn through water Baptism

Happy is our sacrament of water, in that, by washing away the sins of our early blindness, we are set free and admitted into eternal life!

But we, little fishes, after the example of our ΙΧΘΥΣ Jesus Christ, are born in water, nor have we safety in any other way than by permanently abiding in water; so that most monstrous creature, who had no right to teach even sound doctrine, knew full well how to kill the little fishes, by taking them away from the water!

Tertullian
On Baptism
Chapter I

Confession is a second reserve against hell after Baptism

Therefore, since you know that after the first bulwarks of the Lord’s baptism there still remains for you, in exomologesis a second reserve of aid against hell, why do you desert your own salvation? Why are you tardy to approach what you know heals you?

Tertullian
On Repentance
Chapter XII