I've come up with some examples to apply the Hurewicz theorem to compute $H_1(X)$.
This is only interesting if $\pi_1(X)$ is not abelian. The only examples of $X$ such that $\pi_1(X)$ not abelian I can come up with are $\vee_i S^1$ and $\Sigma_g$ the surface of genus $g$ for $g > 1$.
Does anyone know any other examples, preferably easy ones? Many thanks for your help!