Disclaimer: I am not a topologist.
I was taught basic homotopy theory (fundamental group, van Kampen, but not sure about higher homotopy groups, that might have been elsewhere) at the end of a point-set topology graduate course based on Munkres's Topology: a first course. As Mikael comments above, $\pi_1$ being so geometric means it can be taught without the need of the standard algebraic topology machinery. Of course, $\pi_1$ is a far cry from homotopy theory, which requires a lot more technology.
