When Vader said that " I am your Father" Luke just believes it! Without any research, questions etc. Even Ben didn't reply.... Why? Why he did accept it so easily!? Maybe Vader would have lied to him so that he joined him, didn't Luke think that? Are the people in a galaxy far far away so believing?
In Return of the Jedi, Luke asks Yoda flat out if it was true.
Yoda: No! Not yet. One thing remains. Vader. You must confront Vader. Then, only then, a Jedi will you be. And confront him you will.
Luke: Master Yoda...is Darth Vader my father?
Yoda: Er, rest, I need. Yes, rest.
Luke: Yoda, I must know.
Yoda: Your father he is. Told you, did he?
He felt it was true
Although he did confirm it later, he also immediately instinctively knew it was the truth. That's why he reacted so strongly in the moment:
Luke: No...that's not true...that's IMPOSSIBLE!
Vader: Search your feelings, you know it to be true!
Luke, being at least somewhat trained in the Force, was able to 'feel' the truth once his mind was guided to it.
In Jedi, indeed in the scene directly after the Yoda scene, we see Luke again intuiting the truth about his family.
Kenobi: That is the reason why your sister remains safely anonymous.
Luke: Leia. Leia's my sister.
Kenobi: Your insight serves you well. Bury your feelings deep down, Luke. They do you credit, but they could be made to serve the Emperor.
Luke's knowledge of the Force, though far from complete, was quite sufficient for him to find the truth about these claims.