Since inevitably Rudin came up, let me reiterate that although his books are the choice of many instructors they are not favoted so much by most students. I heartily dislike them and do not recommend them at all except to people already knowing the material. I have read two of them and taught from them over the years based on their reputation and always been disappointed both for learning myself and teaching to my students. When I moved recently they did not make the cut to be kept in my personal collection. Books by Sterling K Berberian on the other hand are notoriously user friendly, as well as being also authoritative. The average person can actually learn something from these. George Simmons also writes in a very helpful style. At a more advanced level I actually like books by Jean Dieudonne', although his books are much more difficult. I still prefer them to those of Rudin since at least one feels he is getting some return on his struggles commensurate with the effort from Dieudonne'.
Puzzled by Rudin's fame, I googled a bit and found he actually won the 1993 Steele prize for exposition for those books on analysis that he wrote, but it makes me wonder what the criteria were that were used. Indeed the books are very precise and meticulously organized and correct. But they have essentially no motivational content, and fail entirely to distinguish the important from the trivial, and have as I recall basically no geometric intuition. When I write books I always try to answer the questions of why we are doing what we are doing, and how one might have thought of the solution oneself. These questions, which I consider the most crucial in making a subject understandable and memorable, are apparently never considered in Rudin's writing at all. He seems concerned only with the logical organization of the material.
I have not found much either in the way of references to important research results of his, although his phD thesis was reviewed by someone. I admit readily however that books by outstanding researchers are not always excellent texts for learning.
I apologize for the negative tone of this, but I feel very strongly that Rudin is guilty of making many people feel they cannot learn analysis, or at least that it is inordinately difficult, and I wouldn't want a new student to think that failure to grasp the material as it is presented in Rudin is a sign of his/her own weakness.
On the other hand, if by chance Rudin appeals to you, then go for it. In my opinion you are spending more time and effort than necessary to learn from him, and there will be fundamental points of understanding that are not made clear, but if it suits you, then it is fine. The basic criterion (at least for beginners) is to read whatever speaks clearly to you.
