For simplicial/cellular cohomology, one way to think of it is in terms of dual cell structures: if $a_1,a_2,\dotsc$ are your $k$-simplices or $k$-cells, then they generate the $k$th chain group, and the $k$th cochain group is generated by their duals $\alpha_1,\alpha_2,\dotsc$, where $\alpha_i(a_j)=\delta_{ij}$. You can then draw a dual cell structure which has an $n-k$-cell for every $k$-cell in your original space, with each cell representing a cochain, and with that cochain sending the chains it intersects to $1$ and the rest to $0$ (and extending linearly). So if your space is a surface, you put a vertex inside every face, draw an edge between two of those vertices if there's an edge between the corresponding faces, and add a face between a set of edges if the dual edges all intersect in a vertex.

Then the homology of the new cell structure is the cohomology of the original structure. With field coefficients on a manifold, at least. But it does allow you to visualize, for example, the coboundary map: in the case of our surface, it sends $C^1$ to $C^2$, but $C^2$ "looks 0-dimensional", and so the coboundary map "looks like a boundary map," which is something your students are hopefully familiar with.

Also, if you do it with Platonic polyhedra, you get other Platonic polyhedra. A cube becomes an octahedron, etc. Of course, they all have the same (co)homology but they have different (co)chain groups so they're nice trivial examples, and more interesting than just an arbitrary sphere.

Hatcher goes over this very briefly in the beginning of his chapter on cohomology. He also gives a thing you can do with $\mathbb{Z}$ coefficients that's similar, though in this case you can run into torsion, so I don't know if it's as good an example.

The other thing you can do is just take the no-nonsense algebraic tack. We like cohomology because sometimes we want maps going the wrong direction. For example, its ring structure is easier to work with than homology's coalgebra structure (if your kids are familiar with homology, at this point you show them the coalgebra structure induced by the diagonal map and how it's hard to work with). And you get so much information for free just by knowing that certain maps are ring homomorphisms rather than graded abelian group homomorphisms. I can't think of a good example off the top of my head, but I know there is one.

Oh, I was taught de Rham cohomology before I even knew what homology ways. I think it's pretty easy to understand. That's another option.