I am stuck on the following:
How do I go about finding surjections from the free group of rank 2 $\mathbb{F}_2 = \mathbb{ Z∗Z}$, to the finite group of two elements $\mathbb{Z}_2$.
Also, how would I also prove that this list is complete?
any help on this would be greatly appreciated.
In general, when you're looking for surjections $G\twoheadrightarrow H$, the best thing to do is look at possible kernels $K\unlhd G$, as you will have $G/K\cong H$. These won't always necessarily give you all the information you need, but they can certainly narrow it down. In particular, since $H$ has order $2$, you want to look for normal subgroups of $\mathbb{F}_2$ that have index $2$. As Thomas Andrews suggests, the easiest way is to find them is to use images of the generators. If you are still stuck, see if you can find the normal subgroups, and then you can try to contrive maps of the generators that will give you the right kernels.