How can we detect if a topological space has "holes". I realize that this question might seem ambiguous, is there a topological notion for what a Hole is? I think it has something to do with the fundamental groups of the topological space but I don't really know, could someone please tell me the basics and what is a basic tool for detecting when a topological space has "holes"?

Background: I am currently taking topology 1 and I am halfway through my second semester of abstract algebra.

Thank you kindly, regards.
My professor exposed me to L.S. Pontryagin's "Foundations of Combinatorial Topolology," just recently, and I would say that I had about the same background as you did when I started it. I struggled at some point, but in general I was able to get through it pretty okay. The main interest of the book is simplicial homology, and this will address your question. The text is short, and relatively dense, but it was possible when I had taken a course in abstract algebra and point-set topology.

To be fair, this is probably overkill just to understand a joke about topological holes, but it is a thorough treatment of homology theory.
Homology counts the number of "holes".