I'd like to mention one more thing, which is very important to keep in mind when studying pure mathematics: if you feel like you don't quite understand something, do NOT, under any circumstance, "pressure" or force yourself into believing it. While the saying "practice makes perfect" is perfectly applicable to computation, you have to be very careful when studying pure mathematics to fight the instinct to try and convince yourself that you understand something. It is perfectly okay not to understand something. The chances are, your best shot at understanding something that makes you uneasy is to simply admit your present situation doesn't yield any direct lines of attack toward a better understanding, and to tuck the problem in the back of your mind, until a later time, when further study (possibly in an unrelated area) unexpectedly leads you to the missing piece. You also have to be on guard against allowing trying to form an intuitive understanding of something, before you've fully understood how the idea follows from the definitions and proofs alone. Forming an intuitive understanding of ideas in mathematics is always about understanding the connections between abstract ideas, but the abstract ideas themselves should have no intuitive basis when considered by themselves. For very simple things, it is probably okay to rely on intuition in a pinch (e.g., you can safely think of the derivative of a function at a point as the slope of the line tangent to the graph of the function at the point).

To use a crude analogy, you could think of mathematics as one giant puzzle, but with the pieces coming in slowly, one at a time. Any two pieces have a fairly small chance of fitting together, but since you only have the ability to focus on one or two pieces, you need the memory of old puzzle pieces which previously did not fit anywhere in the back of your mind, so that when you do stumble upon the fitting piece, you can go back to it.

Another thing to keep in mind is that the best truths in mathematics are the most general. Every time you consider a specific example, you should always have some amount of innate desire to see a more all-encompassing idea which handles the details of the specific example as a special case. It will usually not be possible for you to come up with the right generalizations yourself, though; mathematics has evolved gradually over the last couple millennia, and it has taken the trial and error of many brilliant people before the "right" abstractions were found. Your best hope is that your teacher (or author) is leading you on a path that will eventually allow you to see how the things you've come to accept as true can be thought of as existing within a broader framework. The fact that applied mathematics books generally do not do this at all is the reason why one cannot simply try to learn about a topic in applied mathematics, and then try to learn the pure setting as an afterthought. If you go straight to the applications and computations, the chances are that you will be leaving out the conceptual legwork that will be needed to understand the subject in a way that can allow you to potentially create new theory.

To take the puzzle analogy further (to the point of breaking it), you can try to view the generalizations as pieces that connect to MANY puzzle pieces simultaneously (which is obviously not how an actual puzzle works, since each piece only connects to adjacent pieces). As you progress in mathematics, you start to see that the subject is composed of a sort of hidden hierarchy, in which you later learn that your past findings are subsumed by more general theories. Without this, the subject would be unwieldy, since no normal human is capable of committing to memory a perfectly interlocking body of thought that is only made of mostly isolated ideas. Inevitably, you will need some governing ideas, which form the root of a sort of conceptual hierarchy. However, this conceptual hierarchy is more or less impossible to convey pedagogically (c.f. all the complaints about "New Math" back in the `60s), without first understanding all the pieces involved.

(For reasons discussed in the above paragraph, you should be prepared to accumulate a very large number of books and documents, should you begin to more broadly become interested in mathematics).

One way to increase the probability that you'll find interlocking pieces in the same span of attention is to be guided by an excellent teacher (and in some cases, an excellent author). Otherwise, your best shot at exploring the space of possible directions to take is to follow this advice of Paul Halmos: "A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one."

All that said, it is true that any successful student of mathematics will eventually reach a point in his or her studies in which the writing of proofs has become natural enough that, when given a theorem that has a straightforward proof, it the student will probably be able to find it 80% of the time without too much stress or outside help. Getting to that point is important; therefore, a significant chunk of the value of studying a book like Spivak's or Pugh's is to increase your ability to write proofs. This will be a gradual process, so don't be too discouraged when you get frustrated. If you feel like you need to improve your proof writing skills, though, it would certainly help to take a break from the analysis text, and read an elementary book on proofs (just search Amazon or a university library) until you feel like you've done a good job of building up this skill. The ability to write proofs with ease is as important in pure mathematics as algebraic manipulations is important in applied mathematics.

One last warning. If you truly become skilled at pure mathematics, be aware that it can be addictive. Research mathematicians spend their entire lives on this stuff, and are most often quite happy to give up a great deal of things which non-mathematicians value (e.g., a career in industry).

Of course, this really depends on how addictive a personality you have, or if you are unfortunate enough to be a creative genius.