I'll go through the paragraphs of the third edition, motivating why you should/shouldn't consider them
INTRODUCTION
You should carefully read this. This paragraph is not so important for the subsequent development, but it's fundamental for your understanding of the utility of $\mathbb{R}$, it contains an enlightening example wich shows you (with some weird algebraic trick) that $\mathbb{Q}$ contains gaps and so we really need to "patch" it in order to do interesting things.
ORDERED SETS
You should read only the definitions of bound and least upper bound (or supremum). They really matter and are used, explicitly or not, in many theorems. In particular, the latter is subtle and you should practice with it, for example with exercises 4 and 5 at the end of the chapter. The rest of the paragraphs regards ordered fields, you can skip it if you are used to work with $\mathbb{R}$ and its ordering, and this practical experience should suffice to you. Perhaps, you could find strange and interesting that $\mathbb{C}$ cannot be ordered without destroying its algebraic properties.
FIELDS
You can skip this. If your interest is real analysis, your only field will be $\mathbb{R}$ (perhaps $\mathbb{C}$ sometimes) and as said above, the practical properties of field and ordering should be enough for you (e.g. if $a,b,c \in \mathbb{R}$ and $a < b$ then $a + c < b + c$, you cannot divide by zero, etc.)
THE REAL FIELD
There are two ways to build $\mathbb{R}$. The first is axiomatic: you say "how I'd like to work in place that has such property" and magic! You have it by axiom. The second way is contructive: you take $\mathbb{Q}$, do something on it and come up with a mathematical structure that act as $\mathbb{R}$, has the properties of $\mathbb{R}$ and you eventually call it $\mathbb{R}$. It's very subtle and not practically useful, you should skip the latter method, reported in the appendix of the chapter, and know that when you follow the axiomatic method your are speaking of something that exists, in some mathematical sense. You should also consider theorem 1.20 (archimedean property and density of $\mathbb{Q}$ in $\mathbb{R}$) if you don't read the proof, at least carefully read the statement, it's very used and justify some mysterious things as: if $a \in \mathbb{R}$ and $0 \leq a < \epsilon$ for all $\epsilon > 0$ then $a = 0$. Jump over the existence of the n-th root of a positive real, it's intuitive and you can prove it later in different (and simpler) ways.
THE EXTENDED REAL NUMBER SYSTEM
Not only it's not very useful, I think it's dangerous to introduce symbols for infinity when someone still isn't completely conscious of what infinity is and how it acts in many theorems of analysis. Skip.
THE COMPLEX FIELD
As in the case of the real field, if you know what $\mathbb{C}$ is and how to work with it, you can safely skip this, or read it later if you need. The only thing that you probably need are the trianglular inequality in theorem 1.33 and theorem 1.35, known as Cauchy–Schwarz inequality.
EUCLIDEAN SPACES
Read it, it's used in the following chapters.
APPENDIX
As said above, skip it.
EXERCISES
As said above, 4 and 5 are very useful. I also suggest you to work on 6 and 7, they teach you what we mean when we say things like $3^{\pi}$ or when we talk about logarithms.
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.
Best Answer
Principles is an excellent text, but I don't think it's well-suited to self-study. There's nothing wrong with it, honestly, and you'd probably be fine reading it, but to me it's one of those many excellent texts that doesn't really "stick" to the reader that well. It's a better text for an intensive undergrad course with a good professor.
There's a definite distinction between a good text for a lecture course and a good text for self-study. Usually the former contains a broader list of topics and excellent problems, but is very terse and emphasizes logical structure and organization over a readable narrative. It's meant to be studied from, to clarify the structure of the subject to the student and help hammer the points home. Rudin's books fall into this category. For actually reading the material and getting the most benefit from it, I prefer books that take a more classical approach. They tend to have more motivation, examples, and a clear narrative from the author. For this I would recommend Pugh's Real Mathematical Analysis. After this, if you're interested in learning more analysis, I'd recommend Royden. Folland is a great text as well, but falls more under the first category (great reference, good for a lecture test, but I found it difficult to read on my own).
A couple of other alternatives:
Consider reading Stein's analysis series. It's aimed at an undergraduate level, and would be a better place to start than going straight into Rudin if you don't have a good background in analysis. Once you've seen enough analysis, it's not too difficult, but the first encounter can be rather discouraging. (I was a terrible student in my first hard analysis course. I was lazy and didn't put the work in. Expect your first run-in with analysis to be extremely frustrating, but don't get discouraged. With enough work, one day it all "clicks".)
Prof. Su from Harvey Mudd has a first semester analysis course up on YouTube. Harvey Mudd has one of the best undergrad math programs in the world, and it's evidenced by his lucid teaching style. He somehow makes even Rudin easy for students seeing it for the first time.