I'm taking (undergraduate) algebraic topology this year and I have learned some basic concepts in this subject. I found this subject interesting, but it seems like the usefulness of fundamental groups is restricted to low-dimensional.
Pedagogically, it's quite common that the notion of "topological space" is introduced as a metric space at the first time, then introduced later as a general topological space. I hate this approach
Just like general topology, I feel like "fundamental group" is introduced first, then "homotopy group" is introduced later even though the general "homotopy group" have almost all basic properties of fundamental group. Am I correct?
So my question is: can I just jump to homotopy group chapter and then back to first chapters of algebraic topology texts?
The fundamental group and higher homotopy groups are defined only for a space with a chosen base point, a structure which looks like a space, but is a bit more. I explain in this presentation, given in Paris last June, how the problem with the van Kampen theorem and the fundamental group of the circle led me to the use of groupoids and then higher groupoids in algebraic topology.
One has algebraic models of weak homotopy types, and higher van Kampen type theorems for filtered spaces, and for $n$-cubes of spaces, and these allow for some calculations of $n$-types not available by other means. The catch is that calculation of a homotopy group, or a $k$-invariant, from the determination of the model may be tricky!
Whereas group objects in groups are just abelian groups, the situation is quite different for groupoids: $n$-fold groupoids get "more nonabelian" an $n$ increases, and in fact model weak homotopy $n$-types. Since $n$-fold groupoids are strict algebraic structures, one can discuss, for example, limits and colimits. Also the fact that these structures exist, and generalise groups, raises the question of how generally useful they can be.
The work on filtered spaces is covered in this book Nonabelian Algebraic Topology. "The aim is for the major parts of this book to be readable by a graduate student acquainted with general topology, the fundamental group, notions of homotopy, and some basic methods of category theory."
A survey on the $n$-cube case is in this paper.
The $2$-dimensional case, involving pushouts of crossed modules, the latter seen as models of homotopy $2$-types, is given in detail in the above book; this area seems not to have been looked at much by geometric group theorists.
To go back to one part of the question, the work on fundamental groupoids of orbit spaces as "orbit groupoids" given in Chapter 11 of the book Topology and Groupoids has not to my knowledge been lifted to higher dimensions.