My apologies for the rather unorthodox question. I am having some difficulty in finding, as mentioned in the question title, the fundamental group of a generic topological space X. Assume that I know everything in Munkres, Topology 2 ED up to the Seifert-van Kampen theorem (so an undergraduates basic topology course). I understand all the proofs, and the ideas behind them, but am having serious difficulty in methodology in finding the fundamental group given a topological space, and where/how to apply the results studied.
In short, if I have a generic topological space, how would one go about determining its fundamental group? Are there any particular procedures/ideas? How would you go about it? My question is mainly a methodological one.
Thanks to all who reply and I apologise if this is a duplicate, as I can't seem to find an answer to my particular question on the site.
Computing the fundamental group of a space $X$ is in general not easy. There are a few methods which allow you to reduce complexity (for example the Seifert–van Kampen theorem, see the above comments), but you cannot be sure that you will be successful in the end.
However, for a CW-complex $X$ you can us cellular approximation to show that $\pi_1(X) \approx \pi_1(X^{(2)})$, where $X^{(2)}$ is the $2$-skeleton of $X$. This is a substantial simplification.
For finite CW-complexes there exist algorithms to compute $\pi_1(X)$. See for example http://hamilton.nuigalway.ie/preprints/FundGrpAlgo-v10.pdf .
For polyhedra (which are special CW-complexes) you get a presentation of the fundamental group (generators come from $1$-simplices and relations from $2$-simplices). See https://mathoverflow.net/questions/304481/algorithm-for-computing-fundamental-group-of-simplicial-complexes . Unfortunately this is not as good as it looks at first glance. Frequently you will not be able to find out which group is described by this presentation, not even if it is trivial or not.
For more general spaces it seems to me that you have to treat them individually. A nice example is the Hawaiian earring as mentioned in Mees de Vries' comment. This is a really hard job.