I did some work a couple of years ago on phantom maps, but never wrote any of it up because I didn't prove what I wanted. I was focusing on the case of ring spectra. Since it's possible you'd introduce spectra to a class knowledgeable with homology and cohomology, I'll give you a couple of examples from that category. My favorite two papers on this are:
- Phantom Maps and Chromatic Phantom Maps by Hovey and ChristiensenChristensen
- Homological Dimensions of Ring Spectra by Hovey and Lockridge (preprint)
In the first paper they prove, as Proposition 2.5, that all maps from $\hat{S^0}$ to $H\mathbb{Z}$ are phantom. This proof only requires the Universal Coefficient Theorem, a straight-forward computation of $H^0(\hat{S^0})$ and $H^1(\hat{S^0})$, and the fact that Hom$(\hat{\mathbb{Z}}, \mathbb{Z}) = 0$. Even if you waive your hands a bit about what this $\hat{S^0}$ is and tell them it's a completion which has $H_*(\hat{S^0}) = \hat{\mathbb{Z}}$, you'd still have an example here without needing to go much into the theory of spectra.
Corollary 2.7 might also be interesting, but requires more knowledge of spectra: "let $X$ have bounded below cohomotopy groups and let $Y$ be a finite spectrum. Then all maps from $X$ to $Y$ are phantom"
In the second paper, the following wisdom is given (for $E$ an $S$-algebra, aka ring spectrum), which might help for a class seeing this for the first time:
"Recall that a phantom map is a map $f$ for which $[C, f]_∗ = 0$ for every compact $E$-module $C$. So, if we think of a map whose cofiber is a ghost as the homotopy-theoretic analogue of an epimorphism, then a map whose cofiber is a phantom is the homotopy-theoretic analogue of a pure epimorphism"
If you like these examples, I can check through my stack of papers about phantom maps more completely and try to come up with more nice examples. Also, probably Neil Strickland has written something on the subject. He's got lots of notes for various courses on his webpage and seems to like thinking about phantom maps.
