Struggling with Topology. Any advice?

Don't beat yourself up over struggling so much that you doubt your ability to succeed in graduate school. Perseverance and hard work will go a long way. Don't give up. (But yes, consider how to learn.)

Be sure to understand the important things, and don't stress if some of it does not click in your first exposure. (In particular, I don't think the example you mention is central. Grasping continuity, compactness and connectedness is more important. Having a variety of examples that you can work with is also useful. For example, metric spaces are nice. When I'm thinking of a theorem, I have often create subspaces of $\mathbb{R}^2$ in my mind to work with. I suggest you talk with your professor to see what he/she thinks is most important for your class.

Also, do you have the chance to work together with some of your classmates to talk about the material?

Although Munkres' text most certainly has its strengths, from my little exposure to it, I'd guess that I would not have liked my first introduction to the subject to be that text. Rather, I used a book by George Simmons, titled "Introduction to Topology and Modern Analysis." You should try it out when you have time and/or check out some other book.