If you don't believe everything that the fathers said about sources for the gospels, will you please tell me why?
It's not that I don't believe the Fathers, it's rather we have to be really careful about what they said, and how we interpret what they said.
The key figure in this is Papias of Hieropolis (c60-130CE), being the closest to events.
Papias, who knew at least some of the apostles, says that Mark was the
interpreter of Peter’s preaching in Rome, whereas contemporary scholarship, for the most part, states that Mark, like the other Gospels, is anonymous and that the attribution to Mark is a later addition.
What we lack, in all of this, is a chain of evidence. We're dependent upon claims made by exegetes, and there are reasons, scholars assert, to question their reliability ... what we take in faith, however, is a different matter.
In faith I believe Mark listened to Peter's preaching, and used that as the basic material upon which he composed his narrative.
If, therefore, there is a direct link between Peter and Mark, and through Mark to Matthew and Luke, and Luke to Acts, one can see evidence of a 'Petrine tradition', if only because of the prominence which the Synoptics and Acts give Peter, while James, who was a foundational figure in the post-crucifixion community is hardly mentioned at all.
The Johannine materials (Gospel, Epistles and Apocalypse of John) are, according to Papias, attributed to John the Elder, and not the apostle John b. Zebedee. Polycrates agrees with Papias on this point.
Papias on Matthew famously claimed that he wrote a Hebrew Gospel (either Hebrew or Aramaic), a view echoed by ancient authorities, whereas it's generally held that Matthew was composed in Greek and not translated from a Semitic language.
A note here is that Josephus claimed to pen an Aramaic version of
The Jewish War, as well as a Greek one. These are two separate works, not a translation from one tongue into another, but two independent accounts by a bilingual author.
Or it may be that our canonical Matthew is not the Hebrew Gospel Papias is referring to, but a separate text, and that the Hebrew is lost. Some argue a free translation into Greek, others that that when Papias writes "Ἑβραΐδι διαλέκτῳ", he means a
Hebrew style of Greek.
And Q just muddies the waters even further.