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.
"Cte" is an abbreviation for "constante", which means, as you can guess, constant.
Best Answer
Take a book in French that has been translated into English and whose subject matter interests you.
Read a section in French with the translation in English open at the corresponding page and use it if a word or phrase eludes you. Here are a few examples of such translated books:
AUDIN, Géométrie $\iff$ AUDIN, Geometry
DIEUDONNÉ, Fondements de l'analyse moderne $\iff$ Foundations of modern analysis (FREE!) GODEMENT, Analyse mathématique I $\iff$ Analysis I
PERRIN, Géométrie algébrique. Une introduction $\iff$ Algebraic Geometry. An Introduction SAMUEL, Théorie algébrique des nombres $\iff$ Algebraic Theory of Numbers
HERGÉ, Les bijoux de la Castafiore $\iff$The Castafiore Emerald