reference request - How to prove that a map is a Serre fibration?

I want to prove that the homotopy groups of some topological space $B$ of interest to me (not a CW complex) are trivial. I have a strategy of proof that consists in introducing another space $E$ that is contractible, and easily comes with a continuous surjection $pi :Eto B$. If I can prove that any continuous map $f:I^kto B$ lifts to a continuous map $tilde f : I^kto E$, then I'm done.

If I am not mistaken, this lifting property is true as soons as $pi$ is a Serre fibration. Here is my question: are there classical way to prove such a thing, and were can I learn them (or simply learn about Serre fibrations)?
Of course, any reference for the initial problem, which seems slightly weaker, is welcome too.

I guess that I should be able to manage my case by hand, but I think it may be an opportunity to learn more mathematics.

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