A couple people mentioned the Steenrod algebra briefly, but you can do a few more topologically-related things:

• The subalgebras $\mathcal{A}(n)$ of the Steenrod algebra generated by $Sq^1, ..., Sq^n$ are neat. In particular, it is a good exercise in cohomology to compute $Ext_{\mathcal{A}(n)}(k,k)$. (One can do a minimal resolution and try to look for a pattern, and then prove that it works using a spectral sequence.)

• You can show that the Hopf algebras given by $\mathbb{Z}[c_1, c_2, ...]$ ($c_i$ living in degree 2i) and $\mathbb{Z}/2 [w_1, w_2, ...]$ ($w_i$ living in degree i) and comultiplications given by $y_n \mapsto \sum y_i \otimes y_j$ on the generators, are self-dual Hopf algebras and explicitly describe the relationship between itself and the dual. This is neat in and of itself, but then you can mention that these results lead to quick calculations of $H_*MU$ and $H_*MO$ as comodules over the dual of the Steenrod algebra, and thus allow for computations of cobordism groups via the Adams spectral sequence. This self-duality can also be used for a quick proof of the Bott periodicity theorem, though the only reference I know of for this is not yet published (by May), though it will be soon.

• If you're doing cohomology, it's always nice to do the cohomology of an exterior algebra; i.e. a Hopf algebra that's an exterior algebra on primitive generators. It's a very easy result, but you can use it to compute other things via spectral sequences.

I know I've forgotten several things I wanted to mention... if I remember them, I'll edit them in.