Homotopy groups of $S^2$ I'd like to understand higher homotopy groups better and I guess there's no simpler way than understanding them for as simple spaces as possible; therefore $S^2$.
My question essentially has two parts, one geometrical and one computational. First part is not really about spheres per se but I think it should be easier to answer in this context


*

*what do those groups really tell us about $S^2$ and where does all the complexity come from? I mean, naively, I would expect most of the groups to be trivial (as for $S^1$). This probably means that I don't yet have an intuitive grasp of higher homotopy groups; so intuition is what I am looking for in this part.


*have those groups been computed completely (or at least, is there an algorithm to compute them)? I know the theory of higher-dimensional spheres is complicated but perhaps the case of $S^2$ (and therefore also $S^3$) might be a bit simpler.

Regarding 2., wikipedia page mentions that the problem has been reduced to combinatorial group theory of Brunnian braids. Could someone expound on this or provide additional reductions to purely combinatorial problems?

At least at $p=2$, Toda's calculations do not go nearly that far up. 
There are later papers that go a little further, but his  ``Composition
methods in homotopy groups of spheres'' goes up to $n=19$.  It is misleading
to think of $S^2$ as particularly simple.  There is an old theorem (Serre?)
that if $X$ is any simply connected finite CW complex that is not contractible,
such as $S^2$, then for each prime $p$ there are infinitely many $n$ such that
$\pi_n(X)$ has $p$-torsion.
