Maybe every algebraic topology student, at some moment, will ask himself/herself the question: why are $\pi_*$ so difficult and mysterious, especially when compared with (co)homology? Think about the weird connections between $\pi_n(S^k)$ and number theory... it is insane!
But one day I realize this idea may be wrong and biased.
Homology, is probably not really easier than homotopy.
One tends to think $H_*$ is easy, only because most of us only care about finite-dimensional manifolds in our daily life. And then one have that nice vanishing result for higher-dimensional $H_*$.
Consider the infinite-dimensional Eilenberg−MacLane space $K(\mathbb{Z}, n)$ when $n>2$. Its $\pi_*$ is surely as easy as one could hope, but how about its $H_*$, especially the torsions?
It appears to me, the better (?) statement might be
"Spaces with simple $\pi_*$ tends to have complicated $H_*$.
Spaces with simple $H_*$ tends to have complicated $\pi_*$."
This gives one some strange feeling. It is almost like some kind of "Fourier transformation", some kind of duality.
And this makes some spaces particularly interesting: spaces with both simple $H_*$ and simple $\pi_*$ at the same time.
Let me start with some simple examples(products of them give more-trivial examples which shall be ignored here) :
1) $S^1 \simeq K(\mathbb{Z}, 1)$.
2) $\Sigma^g$, that is, Riemann surfaces.
3) $K(\mathbb{Z}, 2) \simeq CP^\infty$.
4) $K(G, 1)$ when $G$ is a finite group.
And I am looking forward to your examples of such spaces, especially comments from AT experts.
Thank you very much.
