So this is what I have for the injectivity:
Knowing $\mathbb{Z}$ is a principal domain, $\mathbb{Q}/\mathbb{Z}$ is an injective module if and only if $\mathbb{Q}/\mathbb{Z}$ is divisible.
Checking the divisibility: Since $\mathbb{Q}/\mathbb{Z}$ is abelian, we want to see if it is divisible as a $\mathbb{Z}-module$. Let $a+\mathbb{Z} \in \mathbb{Q}/\mathbb{Z}$ and $k \in \mathbb{Z}$ with $k \neq 0$. Since $\mathbb{Q}$ is divisible, there exists $x \in \mathbb{Q}$ such that $kx = a$; thus $k(x+\mathbb{Z}) = a+\mathbb{Z}$. Hence $\mathbb{Q}/\mathbb{Z}$ is divisible and therefore INJECTIVE.
Is this correct? For the not projective bit, would it be enough to show that every element in $\mathbb{Q}/\mathbb{Z}$ has finite order?