Here are some hints. Let me know if I should say more.
For the first map, we know that $M$ maps onto $M/\mathfrak{m}M$, which is a vector space over the residue field $A/\mathfrak{m}$, so it should be enough to show that $M \neq \mathfrak{m}M$. Do you know techniques for doing this in the local case?
For the second, you need to find an $x \in M$ such that $\operatorname{ann}(x) = \mathfrak{m}$. One way of doing this is to consider the family of ideals $ \{\operatorname{ann}(y) : y \in M,\, y \neq 0\}. $ Show that a maximal element — with respect to inclusion — of this family is prime. Why does a maximal element exist? How does this help us?