I'm studying fundamental groups in Kosniowski's First Course in Algebraic Topology, and I'm really having trouble understanding why fundamental groups are not always trivial. I have taken a knot theory class in the past, which I think is messing with me now, because in my mind every loop based at the same point is equivalent.
My professor has explained to me that on something like a torus, the hole in the center interrupts the action of "pulling" the loop in, so you end up with at least two equivalence classes (I think there should be three?). But in general, I don't really understand how to identify when a fundamental group is not trivial. And given a space like $X=\{a,b,c\}$ with topology $\{\emptyset, X, \{a\}\}$, where do I start with trying to understand the fundamental group, or is that question irrelevant without more information?
Yes, a good way to think about it as having obstructions that don't let you pull your loop back into its basepoint. More rigorously, the following result holds:
Let $X$ be a topological space. $\pi_1(X, x_0)$ is trivial if and only if every loop $\gamma : S^1 \rightarrow X$ can be continuously extended into a function $f: B \rightarrow X$, where $B$ is the unit disk in $\mathbb{R}^2$.
Intuitively, this means that the fundamental group is trivial iff every loop can be "filled" up with a disk. The idea of filling up with a disk indicates some notion of nothing obstructing the interior of the shape drawn by the loop. In more exotic settings it doesn't necessarily have to be a hole which is causing this obstruction; for instance in $\mathbb{R}P^2$ it's more that you can't "pull back" into smaller circles sharing the same basepoint. This is different from a simple hole obstruction and indeed $\mathbb{R}P^2$'s fundamental group is of a different flavor as well (It is $\mathbb{Z}_2$). Here it's sort of like we can't pull the loop back in without "snapping" the "fabric" of the loop.
Later on, there will be better tools to understand and help you calculate the fundamental group such as covering spaces and van kampen theorem.