As others have said, it really depends on what your interests are. There are dozens of directions you could go, each of which can consume more than a lifetime of work and study. I don't know what your situation is, but at some point one leaves the textbooks behind (mostly, at least) and begin to focus on research papers (both contemporary and older), as very little of the enormous research literature actually makes it into monographs and treatises, to say nothing of textbooks. For example, almost none of the various results dealt with in the references I gave yesterday in my answer to $\alpha$-derivative (concept) can be found in any books. (The only thing I can think of off-hand is the Auerbach/Banach paper, whose results I believe can be found in Eduard Cech's 1969 text Point sets.) Of course, much of this research literature consists of tangled paths probably few would want to follow anyway . . .
That said, if for whatever reason you wish to devote two or three years going through a textbook/monograph, I suggest one of the following two:
Nelson Dunford and Jacob Schwartz's multi-volume series Linear Operators
Zygmund's Trigonometric Series
Each of these is a classic and each contains a huge amount of mathematics. Probably the Dunford/Schwartz series is the better fit for you, I suspect, as it has a large number of carefully thought out exercises.
Depending on your interests, however, any of the following should also work, along with dozens of other paths I (or others) could easily come up with.
Pertti Mattila's Geometry of Sets and Measures in Euclidean Spaces, perhaps followed by Herbert Federer's Geometric Measure Theory
Lindenstrauss/Tzafriri's 2-volume work Classical Banach Spaces, perhaps followed by Lindenstrauss/Preiss/Tišer's Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces
Gilbarg/Trudinger's Elliptic Partial Differential Equations of Second Order, perhaps followed by Heinonen/Kilpeläinen/Martio's Nonlinear Potential Theory of Degenerate Elliptic Equations
Steven Krantz's Function Theory of Several Complex Variables
I have a hard time avoiding blatant self-promotion here...
I don't know Lang. Ahlfors is of course a classic. I have a lot of issues with Conway. (My complaints are with the first volume, which it turns out he wrote as a student! The second volume is full of great stuff.) Conway was the standard text here for years - I hated it so much I started using my own notes instead, which eventually became Complex Made Simple (oops. Well, there are things in there that are not in any other elementary text that I know of.)
Two examples that spring to mind regarding Conway:
He spends almost a page using the power series for $\log(1+z)$ to show that $\lim_{z\to0}\frac{\log(1+z)}{z}=1,$ evidently not recalling the definition of the derivative.
There's a chapter or at least a section on the Perron solution to the Dirichlet problem. There's an exercise, like the first or second exercise in the chapter, which a few decades ago I was unable to do. I sent him a letter explaining why it was harder than he seemed to think.
In the next edition the words "This exercise is hard" were added. A year or so later I realized the exercise was not just hard, it was impossible. Asks us to prove something false.
Seems very unimpressive - I complain I don't know how to do the exercise and he doesn't even bother to make sure it's correct.
Best Answer
Principles, the so-called baby Rudin, is undergraduate introduction to analysis. Basic proof writing, continuity, derivative, integral. It has some advanced topics which normally aren't covered in an undergraduate course such as a rigorous introduction to differential forms and a crash course on measure theory. It was meant to be covered in a two semester sequence by mathematics undergraduates. The downside of the book is the lack of any pictures. The story I heard was that Walter had lots and lots of pictures, but at the time the publisher said they'd print no pictures as it would be too expensive to typeset, so they all got removed.
The other book, sometimes called big Rudin is graduate level analysis. It combines what are usually the graduate level real and complex analysis classes that phd students take in their first or second year.