According to Introduction to Topological Manifolds (John M. Lee, p. 298), one can easily build the universal cover of a given topological space $X$ by picking a point $x_0$ at random, and then considering all the possible paths from this base point to any other point in the space. If we call this set $A(X; x_0)$, then our universal covering space would be $A(X; x_0)/\sim$, where the symbol $\sim$ stands for the path equivalence relation.
But my question is: how do you actually use this construction in practice? The proof of this theorem relies on giving the covering space a topology in a pretty unintuitive way, and I'm afraid that's the trickiest part when trying to make use of it.
Suppose I want to find the universal covering space of $S^1$ (forgetting for a moment that we already know it's $\mathbb{R}$): if I choose a point $x_0\in S^1$, then I can say that any two paths that start from $x_0$ and end at different points on the circle cannot be equivalent, and the same is true for any two paths that end at the same point but after having gone around the circle a different number of times. This gives me a rough idea that the covering space must have "the same number of points" as $\mathbb{R}$... but how do I show it's $\mathbb{R}$ using solely this construction?