After just finally getting a somewhat intuitive grasp on just what the fundamental group of a topological is, I'm curious as to how one would relate the algebraic properties of the group to the topological properties of the space. For instance, in an annulus, one has the identity loop, and a loop which wraps around the "hole" in the space, and all multiples and inverses thereof, obtaining the infinite cyclic group. So what it looks like is that the fact that we have this single cyclic group being generated by one loop is telling us that there is precisely one classic "hole" that can be "looped around" any number of times. Similarly, how can one infer other, more exotic types of "holes" in the space from the fundamental group?
An example that, to me, is a little spooky is that the real projective plane has fundamental group isomorphic to $\mathbb Z_2$. So, essentially, there is a sort of strange hole around which one can wrap a loop, but if you wrap around it twice, the loop contracts back to nothing. And I have no idea what it might mean intuitively for a space to have a fundamental group that is non-abelian.
I'm also interested in the higher homotopy groups $\pi_n$ of higher dimensional spaces, and what other kinds of strange holes might exist. I might add that I was imagining that something like $\pi_3(S^2)$ couldn't even exist, since it seems it shouldn't be possible to embed a 3-sphere into a 2-sphere, but now I've seen a cursory explanation of Hopf fibrations, and apparently topology is even more bizarre than it seems. I don't really feel ready to think about that one quite yet.
I don't actually have much rigorous knowledge of topology, aside from things like the basic axioms of a topologial space, definitions of continuity, compactness, etc., so I would greatly appreciate a special emphasis on the intuitive concepts that the fundamental group and the other homotopy groups demonstrate.