Find all of the distinct left cosets of <4> in Z18 and all the cosets of <4> in the subgroup <2> of Z18.
So The distinct left cosets of <4> in Z18 are
0 + <4> and 1 + <4>.
Do I have to list all 17 cosets in Z18 or since they are repetitive, the two I have listed cover everything?
Now, all of the cosets of <4> in the subgroup <2> of Z18 is just <2> because <2>=<4>={0,2,4,6,8,10,12,14,16}.
You are correct that the left cosets of $\langle 4\rangle \in \mathbb Z_{18}$ are given by $\langle 4\rangle$ and $1 + \langle 4\rangle$.
In the second case, the number of cosets of $\langle 4\rangle$ in $\langle 2\rangle$ is one: namely $\langle 4 \rangle = \langle 2\rangle$.
Note that in abelian groups, we can speak of the number of cosets, because left and right cosets are indistinguishable.