The Lord Jesus Christ Himself told us as to which Name we are to be baptized into. It is implied in Matthew 28:19 that it is the Name of God (the triune Godhead).
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost.
All performed water baptism references indicate the name of Jesus was used. Also, Jesus opened the disciples understanding prior to His ascension:
Luke 24:45-47
Then opened he their understanding, that they might understand the scriptures,
And said unto them, Thus it is written, and thus it behoved Christ to suffer, and to rise from the dead the third day:
And that repentance and remission of sins should be preached in his name among all nations,
beginning at Jerusalem.
Peter gave specific instructions regarding when and how sins were remitted while in Jerusalem on the Day of Pentecost:
Acts 2:38-39 (Jewish)
Then Peter said unto them, Repent, and be baptized every one of you in the
name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.
For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call.
Then they that gladly received his word were baptized: and the same day there were added unto them about three thousand souls.
Acts 8:12-17 (Samaritans)
But when they believed Philip preaching the things concerning the kingdom of God, and the name of Jesus Christ, they were baptized, both men and women.
Then Simon himself believed also: and when he was baptized, he continued with Philip, and wondered, beholding the miracles and signs which were done.
Now when the apostles which were at Jerusalem heard that Samaria had received the word of God, they sent unto them Peter and John:
Who, when they were come down, prayed for them, that they might receive the Holy Ghost:
(For as yet he was fallen upon none of them: only
they were baptized in the name of the Lord Jesus.)
Then laid they their hands on them, and they received the Holy Ghost.
Acts 10:44-48 (Gentiles)
While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.
And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.
For they heard them speak with tongues, and magnify God. Then answered Peter,
Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
And he commanded them to be
baptized in the name of the Lord. Then prayed they him to tarry certain days.
Acts 11:13-14
And he shewed us how he had seen an angel in his house, which stood and said unto him, Send men to Joppa, and call for Simon, whose surname is Peter;
Who shall tell thee words, whereby thou and all thy house shall be saved.
Acts 19:2-5 (Disciples who did not realize they had to water baptized in Jesus name)
He said unto them, Have ye received the Holy Ghost since ye believed? And they said unto him, We have not so much as heard whether there be any Holy Ghost.
And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism.
Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus.
When they heard this, they were
baptized in the name of the Lord Jesus.