Rudin is a good breviary rehearsal if one already almost-knows and feels the material. People with certain inclinations may get a warm fuzzy feeling how things neatly fit together as if by omniscient design. Otherwise it is a crossword puzzle to amuse the god himself.

You may get a feeling you understood things (and earned that), but you are wrong. When I ask people what they really remember Rudin from, what specific piece of proof they learned specifically from his main and baby books, I hear only vague answers. There is a pretense of “teaching to think” by omission, also common in some dated textbooks, but I personally would leave that to professionals over at philosophy dept. and focus on clear exposition leaving neatness for examples. OTOH if you are predisposed to lauding yourself for how smart you (or Rudin) are for figuring all the tricks (knowledge of which stays at this level), you are in for ego boost (or bust).

Especially for real analysis his treatment of Lebesgue integral is worse than just about anyone else's I know (also, in 21st century it's time for better integrals like e.g. Henstock–Kurzweil). The only thing worse is again Rudin's own treatment of differential forms.

In professional mathematics Rudin is renowed for numerous many things, among them the Rudin–Keisler order in the theory of ultrafilters and ultraproducts. It is a sign about reading order. Because as it happens Keisler also wrote a textbook, and from a diametrically opposed perspective. Equally far fetched in the other direction, one of intuitionistic non-standard infinitesimals. I think being a product of certain totalising era of uniformisation in mathematics these texts are complimentary.

For a reader interested in somewhat extended real analysis I would recommend Lang, Bressoud, Körner („Companion…”).

Mentioned Terence Tao book from weblog-notes for his original RA course is also freely downloadable as pdf.

Finally Strichartz is overly chatty wordier antithesis to Rudin.

The introductory one is known as Baby Rudin and is a pretty good first exposure to real analysis.

This is how I learned it! (Baby Rudin, anyway.)

I wonder if this is still the go-to for undergraduate classes --- does anyone know?

+1 I am currently reading this. It’s dense but very clean.