Let $p: \bar{X} \rightarrow X$ be a covering space projection. Let $[f] \in \pi_{1}(X, x_{0})$. We can define the following map
$r: \pi_{1}(X, x_{0}) \rightarrow p^{-1}(x_{0})$
$r[f] = \bar{f}(1)$ where $\bar{f}: [0,1] \rightarrow \bar{X}$ is the unique lift of f, a loop in $X$.
This map provides an pretty clear isomorphism between $\pi_{1}(S^{1})$ and $\Bbb{Z}$. Moreover, we can prove some properties about this map, such as it being surjective if $\bar{X}$ is path connected, and being bijective if $\bar{X}$ is simply connected. It seems like this could be a helpful tool in calculating other fundamental groups—is that the case? Here is another example:
Example: We have the the quotient map for $\frac{S^n}{\sim}$ where $\sim$ identifies antipodal points is also a covering space map. $\frac{S^{n}}{\sim}$ is $\Bbb{RP}^{n}$, so our covering map looks like $p: S^{n} \rightarrow \Bbb{RP}^{n}$ with $p^{-1}(x_{0})$ having exactly 2 elements (that is how we identified them). Since $S^n$ is simply connected, we have that the map $r$ defined above is bijective. There is a unique group structure on 2 elements, so $\pi_{1}(\Bbb{RP}^{n}, x_{0}) = \Bbb{Z}_{2}$.
In deed it is sometimes useful to use the Galois correspondence for fundamental groups. It roughly says:
If $p\colon (\tilde X,\tilde x_0)\to (X,x_0)$ is the universal covering, then there is an isomorphism between the fundamental group and the group of deck transformations $\pi_1(X,x_0)\to Aut(p)$.
A deck transformation of $p\colon (\tilde X,\tilde x_0)\to (X,x_0)$ is a homeomorphism $f\colon \tilde X\to\tilde X$, s.t. $p \circ f = p$. The set of all deck transformations of $p$ is called group of deck transformations $Aut(p)$. A covering space $(\tilde X, \tilde x_0)$ is called universal covering space, if it is a path-connected and simply-connected space.
The Galois correspondence says more:
There is a correspondance between conjugacy classes of subgroups of $\pi_1(X,x_0)$ and coverings of $X$ without basepoints up to iso. We always need spaces, which are path-connected, locally path-connected and semi-locally simply connected. For more information just search for Galois correspondance and fundamental groups.
In my opinion a more powerful tool to just calculate something is the theorem of Seifert-van Kampen.