Q/Z
$\mathbb{Q}/\mathbb{Z}$ is a pretty terrible abelian group, or a rather hard one, there may be better injective resolutions to work with. It would certainly be easier to do the projective resolution, use $0 \to Z \mathbb{Z} \to Z \mathbb{Z} \to Z/n \mathbb{Z}/n \to 0$. this will surely be easier to work through than the one involving Q/Z. $\mathbb{Q}/\mathbb{Z}$. Then compute the appropriate tensor product or hom group.
I started learning this stuff on more interesting modules as Schmidt suggests. For example, modules over the group ring of some cyclic group, or maybe an exterior algebra on two generators (if you make the generators of different gradings, in particular 1 and 3). This happens to be the category of modules you need to understand in order to compute complex connective k-theory!
this should help get you going. These computations are very fun!
