This statement was recalled by my instructor as general fact in my class of Topology but it doesn't looks trivial to me at all.
Statement: Every continuous map $f : X \to Y$ induces a morphism of groups $\pi_1(f) : \pi_1(X,x) \to \pi_1(Y,f(x))$ and if f is a homotopy equivalence, then $\pi_1(f,x)$ is an isomorphism.
I was wondering if you can give me reference for a proof of both of these statements.
I think I will not be able to figure them out by myself.
As @Randall says, this is in any book which discusses the fundamental group. For example: the first part is in Hatcher's book Algebraic Topology: there is a subsection called "Induced Homomorphisms" starting on p. 34. The second part: Proposition 1.18 in Hatcher.
But you should try to figure them out for yourself first.