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
Best Answer
Rudin's Principles of Mathematical Analysis is a hard book, but it's also a standard and it is extremely well-written (in my and many others' opinion) so you should read it early on. It may not be your first book in analysis, but if not I would make it your second.
When working through Rudin, even though you have a solutions manual, you should not give up on problems before you have solved them. There are problems in that book that take some of the best students hours over days to solve. The process of banging your head against the wall (or the book, or any other hard object) is part of the book and part of your preparation for mathematics. When you do get through Rudin, you will be in a very good place to step into the field of analysis $-$ possibly even the next Rudin book, Real and Complex Analysis.
As a soft introduction to analysis before Rudin, I would recommend my teacher's book: Mikusinski's An Introduction to Analysis: From Number to Integral. It is fairly short and easy to get through, and will prime your brain for the more intense fare of Rudin's book. It only covers single-variable analysis, however, which is 8 out of 11 chapters in Rudin; most courses in analysis only necessarily cover the first 7 chapters anyways.