I'd definitely recommend baby Rudin for general introductory analysis, his followup textbook is also my favourite analysis book. Fourier analysis is generally very reliant on Lebesgue integration. A book which uses only the Riemann integral (if I recall correctly) is Dietmar's first.
The route I took into harmonic analysis was starting with A (Terse) Introduction to Lebesgue Integration by Franks, which has an online draft here, which introduces the Lebesgue measure/integral in a very rigorous manner before establishing the basic $L^{2}$ treatment of Fourier series on $\Bbb{T}$. After that, the best recommendation that most people interested in harmonic analysis should read is katznelson's book, which covers the standard Fourier transform on $\Bbb{R}$ material very nicely, as well as sketching the locally compact abelian group stuff. From there, there seems to be less of a general consensus. I found Rudin's Fourier Analysis on Groups excellent for the locally compact abelian case, giving nice proofs of several theorems for which I wasn't happy with the proofs given in other books. I also enjoyed Classical Harmonic Analysis and Locally Compact Groups by Reiter and Stegeman as more of a broad introduction to abelian harmonic analysis, although it does ommit quite a few key proofs. I can't offer many references beyond the abelian case, and certainly not beyond the compact case, but the second of Deitmar's books was my favourite general reference for introductory nonabelian harmonic analysis. I've not read much of it, but my favourite treatment of the compact case is that found in Folland's book, which is online here; in particular, I found that its description of the representation theory was much more natural than other treatments.
I'm not American, so I can't relate what I've said to the courses you've listed, but hopefully this will help somewhat. I'd certainly recommend starting with Franks and Katznelson.
As a general aside related to your comments, I'd recommend trying to read as much as possible without asking for help from your professors - even if you ultimately have to ask for some help understanding something, you'll get much, much more from it if you only ask for help once you've beaten your brains out trying to understand it.
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.
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Maybe, it would be better for you to choose a slightly more advanced topic (like introduction to topological manifolds using Lee's book, or Boothy's book) instead of multi-variable analysis for next semester. You should be able to cover multi-variable mathematical analysis yourself by using a suitable textbook (like Zorich's Mathematical Analysis II, which covered some more material).
I think the point for independent study is to have a faculty guide you through the learning process. So if the material is not too difficult you can cover it yourself instead of following the instructor. And you can always ask questions at here.