For some reason, I can't find a reference for $\pi_i GL(n,\mathbb C)$ nor can I figure what they are. For most Lie groups, you can get a nice fibration and use the long exact sequence in homotopy to inductively compute the homotopy groups (e.g. the fibration $SO(n-1) \to SO(n) \to S^{n-1}$). However, I can't think of a nice fibration; $GL(n)$ acts transitively on $\mathbb C^n$ but I don't know a nice description for the stabilizer subgroup.
This is motivated by understanding the statement that $GL(n)/GL(k)$ is $k-1$ connected (for the real and complex cases), so if there's an easy explanation for that without appealing to $\pi_1 GL(n)$, then that would also be appreciated.
There's a fibration
$$GL(n, \mathbb C) \to GL(n+1, \mathbb C) \to \mathbb C^{n+1} \setminus \{0\}$$
By Gram-Schmidt, this fibration is fibre homotopy-equivalent to
$$U_n \to U_{n+1} \to S^{2n+1}$$
given by only remembering the 1st vector in the matrix, just as in your $SO(n)$ example.
The stable homotopy groups of the unitary groups are known. Google "Bott Periodicity". The unstable groups for $U_n$, just like for $SO_n$, are only known in a range.
I believe these fibrations are discussed in Bredon's book, as well as May, among others. This is example 4.55 in Section 4.2 of Hatcher's book.