I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
1) Introduction to real analysis by Bartle and Sherbert
2) Methods of Real Analysis by R.R. Goldberg
3) Mathematical Analysis by Tom Apostol
4) Real and Abstract Analysis by Karl Stromberg.
5) A radical approach to real analysis by David M Bressoud by MAA.
The first book is a very good book for a beginner. The next two are classics. (4) is also very good in case you want to read something advanced. The last one keeps entertaining you with some interesting examples as well as some interesting history of Real Analysis.
Gelbaum and Olmsted, Counterexamples in Analysis.
The first real analysis/advanced calculus class is full of theorems with multiple conditions, and it can be difficult to tell which ones are necessary for what parts of the theorem. This book will provide examples for why the theorems are as they are and not otherwise.
There is a set of notes and additional exercises due to George Bergman. See his web page...
Okay, so as you haven't accepted any of the answer, I thought maybe I should give a try to answer as best as I can.
I'm currently doing Chapter-3 from Walter Rudin's text, and simultaneously following these wonderful notes, https://www.math.ucdavis.edu/~emsilvia/math127/math127.html by UC,Davis.
They include proof of each and every theorem and lemma in the book, and even a statement that is left in the book for the readers to prove is proved in these notes.
If you want, you can follow only these notes and not do theory from the book at all, and directly come back to it to solve problems at the end.