TITLE: Examples of fundamental groups
QUESTION [22 upvotes]: I'm starting to study fundamental groups and I didn't find in the books of Algebraic Topology many examples of them. Can you list the examples you know and the demonstrations?
I think it would be useful to a self-student and also to a teacher/professor who can use this in their classes.
What's the most interesting examples do you have in mind? and the most important and basic ones which we can find others fundamental groups?
We can also think about examples which we can use different tools and techniques to training our ability and creativity in discover those fundamental groups. 
Thanks

REPLY [4 votes]: Two earlier  answers omit the study  of the fundamental group of an orbit space. 
In Topology and Groupoids it is shown in Chapter 11 that  for a discontinuous action of a discrete group on a Hausdorff space $X$ (e.g. $G$ is finite) satisfying some good local properties (e.g. $X$ has a universal cover), then the fundamental groupoid of the orbit space $X/G$ is isomorphic to the orbit groupoid of the fundamental groupoid of $X$. So one needs techniques for calculating orbit groupoids, and their vertex groups. These, due to Higgins and Taylor, are given in Chapter 11. As an example, the fundamental group of the symmetric square of a space $X$ is calculated to be the fundamental group of $X$ made abelian. 
Notice the key point that if $G$ operates on a space $X$ then it operates on the fundamental groupoid of $X$, but not necessarily on any fundamental group. 
It is also useful to generalise the theory of covering spaces to non free actions.  Ross Geoghegan in his 1986 review (MR0760769) of two papers by M.A. Armstrong on the fundamental groups of orbit spaces wrote: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." However Armstrong does not use groupoids, which makes his results seem more technical.