How is Loomis and Sternberg's "Advanced Calculus" as an introductory analysis text? Will it be a good substitute for a standard mathematical analysis text like Baby Rudin or should I buy both of them to avoid any gaps in my knowledge before progressing any further?
(P.S., what exactly does "advanced calculus" mean? Every single book I've seen on "advanced calculus" seems to have vastly differing table of contents.)
 A: In the U.S., "Advanced Calculus" usually means the course in university where vector calculus and line and surface integrals are first encountered. For the vast majority of students, it's in a non-rigorous form, but occasionally it may be done rigorously. On the other hand, the "mathematical analysis" course is always rigorous, but tends to focus on more basic questions.
I'm not very familiar with the book by Loomis, so what follows is based on an examination of its preface and table of contents.
First of all, the book is rigorous.
Other than the fact that the book discusses metric spaces, it has very little in common with Rudin's book, however. It presumes that you've mastered rigorous single-variable calculus, and focuses attention only on multivariable questions. It covers in much greater detail the multivariable topics in Chapters 9 and 10 of Rudin, which are considered the weakest parts of that book.
So the question of whether you can skip Rudin depends partly on what your background in single-variable calculus is. If you've learned calculus from a book like Spivak's or Apostol's (with difficult exercises, including ones focusing on theory), you probably don't need to read Rudin, since, for single-variable topics,  Rudin differs from those texts mainly in its more difficult exercises and its use of topology, and you'll be getting that from the Loomis book.
On the other hand, if your single-variable calculus didn't focus much on theory, then it doesn't appear the book by Loomis can substitute for Rudin on topics such as series, aspects of continuity and differentiability that are specific to the single-variable case (for example, monotonic functions and convex functions), and sequences and series of functions.
On the other hand, most of the content in Loomis is either not addressed in Rudin's book, or not treated properly.
