First, I know that any compact connected surface is either $\mathbb S^2$, $\mathbb P^2 \# \cdots \# \mathbb P^2$ ($n$ copies), or $\mathbb T^2 \# \cdots \# \mathbb T^2$ ($n$ copies).
Also, suppose I know the following result:
-Given $X$ is a connected space with a contractible universal covering space, $f : Y \to X$ is null homotopic if and only if the induced homomorphism $f_* : \pi_1(Y, y) \to \pi_1(X, f(y))$ is trivial for each $y \in Y$.
For the first case, I know that since the fundamental group of $\mathbb S^2$ is trivial, it must be that any continuous $f$ from $\mathbb S^2$ to $\mathbb S^1$ should be trivial.
For the second case, when $n$ is just one, since the fundamental group of the projective plane is just a group of order $2$.
However, I am not sure how to proceed with other cases, and show that there exists/does not exist non-null homotopic maps.
Also, what should I do to prove the fact I am assuming?