Suggestions on Real, Fourier Analysis textbooks?
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.
This book doesn't get the recognition it deserves:
Pugh's "Real Math. Analysis."
It gives a very intuitive approach and quit thorough in its presentation.There are many examples, many, many problems including some from Berkeley pre-lim exams.
Especially good for self-study.
I have all 4 Stein and Shakarchi books, and I've found them to be very useful in my first couple years of grad school. My graduate real analysis course used book III, so I bought the others to have the complete set. They have pros and cons, of course. Pros: they cover a ton of material, have more good exercises than you could possibly finish in a year (if you can, props to you), and approach everything in as rigorous a manner as possible. In particular Book I is possibly the most digestible rigorous introduction to Fourier analysis that I know of, at least at the advanced undergraduate level. The first chapter reviews motivations from partial differential equations, but from there on out you're developing the basic Fourier series convergence results and the Fourier transform on $\mathbb{R}$ and $\mathbb{R}^d$. There are tons of 'applications' which is good to stay grounded. The last two chapters are a nice short introduction to finite Fourier analysis and some analytic number theory, a nice contrast to most of the rest of the book.
Book III is again, in my opinion, a very digestible introduction to Lebesgue theory. Whereas many books on real analysis (like Papa Rudin) tend to start with the definition of a measure space, S&S stick with Lebesgue measure and integration on $\mathbb{R}^d$ for most of the book. I would definitely recommend the first 3 chapters. You can probably do better for the abstract measure theory (Papa Rudin is a good choice). Cons of the S&S books: some of the proofs are a bit hasty, so make sure you flush out all the details for yourself. Also, some of the exercises are a worded bit confusing at first. The other problem I have with the books is that each book isn't really self contained - they make a lot of references to the other books in the series.
Another good, albeit lighter, introduction to measure theory is "Measure, integral and probability" by Capinski and Kopp.
As far as getting in to harmonic analysis, I'm not sure if there is a perfect book at your level. I did an independent study course last year and we worked out of "real variable methods in harmonic analysis" by Torchinsky. It was dense, but I was able to get through the basic chapters (Weak Lebesgue spaces, Interpolation of Lebesgue spaces, the Hilbert transform) in less than 10 weeks. Plus it's a Dover book, so it's cheap.
Some might disagree with me on this, but I've found that a great way to motivate what I need to learn is to keep a "goal" book or paper in mind that I'd eventually like to understand. I've had a few of these shelf-dwellers staring me down for the past year or two, including the harmonic analysis Bible (Stein's "Harmonic Analysis"), Grafako's "Modern Fourier Analysis," and so on. Occasionally you might head to the library and just thumb through a book to see where you're at (if you don't recognize most of the words, you've got a lot more work to do!)