I am trying to prove or disprove with a counterexample the following statements:
(i) From every set of generators of a module $M$ one can extract a basis.
(ii) Every linearly independent subset of a module $M$ can be extended to a basis.
I am pretty sure none of them is true since if this was the case, then every module $M$ would be a free module however I can't think of any counterexamples, any suggestions would be appreciated.
As you say, take any non-free module, for example $\mathbb{Z}/2$ over $\mathbb{Z}$. But even for free modules the claims (i) and (ii) are wrong.
(i) $\{2,3\}$ is a generating set of the $\mathbb{Z}$-module $\mathbb{Z}$, but it does not contain a basis.
(ii) $\{2\}$ is a linearly independent subset of the $\mathbb{Z}$-module $\mathbb{Z}$, but it cannot be extended to a basis (since this would have to cardinality $1$, but $\{2\}$ is no basis.)