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.
Based on the comments, I would say you could consider doing a few things before your course:
1) Read Chapter 2 of Royden-Fitzpatrick
2) Read the first chapter of Friedman's Foundations of Modern Analysis, up to and including section 1.6. A very elegant approach to abstract measure theory with a number of concepts (i.e. limsups of sets) that are usually neglected nowadays. Also, a much more gentle intro to abstract measure theory than Rudin (e.g. no complex numbers to deal with).
3) Learn about measurable functions and Lebesgue integration from Bass Real Analysis for Graduate Students (Chapter 5, 6, 7) or Friedman's book (Chapter 2, up to and including Section 2.10).
4) If you're still alive at this point, try reading Chapters 4, 5, 6 of Royden. (By alive, I just mean you may not get this far, and that's OK.)
You won't have seen the Riesz representation theorem up to this point, but you will know the basics of Lebesgue integration. For reference, (1-3) is roughly what I did to learn basic measure theory the first time, and (4) was part of the grad course I took as an undergrad. (I didn't take the grad course as a PhD student.)
Stein and Shakarchi (Measure Theory, Hilbert Spaces, etc.) would be similar to Chapters 2-6 of Royden, but its exercises are harder (and not in a good way). The fact that it doesn't do "abstract" stuff until the end doesn't recommend it either. (Incidentally, it has nice extra material on geometric measure theory at the end, but that's a different story.)
Best Answer
Possibly Abbott, Understanding Analysis