When did Christians start teaching the Trinity? Was it invented at Nicaea?

Didache c. 100 ADDidache 7.1

Now about baptism, baptize in this way: after saying all these things beforehand, baptize in the name of the Father and of the Son and of the Holy Spirit, in running water.

The earliest church manual, about a generation before Justin. The threefold name of Matthew 28:19 is already the fixed baptismal form; where running water is lacking, water is poured three times on the head (7.3).

Justin Martyr c. 100–c. 165 ADFirst Apology 61.3

Then they are brought by us to where there is water, and are reborn in the same way of rebirth by which we ourselves were reborn. For in the name of God, the Father and Lord of all, and of our Savior Jesus Christ, and of the Holy Spirit, they then receive the washing in the water.

The threefold name of Matthew 28:19 used at every baptism. Christians were baptized into Father, Son and Spirit long before the doctrine had technical words.

Theophilus of Antioch fl. c. 180 ADTo Autolycus 2.15

In the same way the three days before the lights were made are types of the Trinity: of God, and of his Word, and of his Wisdom.

The earliest surviving use of the word "Trinity" (Greek trias). Theophilus calls the Spirit "Wisdom".

Tertullian c. 155–c. 220 ADAgainst Praxeas 2.4

...as if, in this way too, one were not all, since all are from one, through unity of substance, while the mystery of the economy is still kept, which arranges the unity into a Trinity, setting out three, Father, Son and Spirit: three not in standing but in sequence, not in substance but in form, not in power but in aspect; yet of one substance, one standing and one power, because there is one God, from whom these sequences and forms and aspects are reckoned in the name of the Father and of the Son and of the Holy Spirit.

The first use of the Latin trinitas and of the formula "one substance", against a teacher who said Father and Son were the same person.