Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to S^2$. It did not go well. I decided I didn't understand $\mathrm{Tor}_{*,*}$ at all, so I asked an algebraist. We went through the computation and successfully proved that $H^*(S^1)$ is infinite-dimensional. That was dispiriting.
So I seem to need some help. Where can I see something like this worked out to get the gist of how to use this sequence?