i think dieudonne's analyis book you mention is one of the best books in existence. the amount of information and the depth to which it goes is virtually unmatched in other works. note however he says in the introduction that it is clear one should not approach his book without first having a firm grasp of classical analysis, which means perhaps a thorough grounding in rudin's work or a more example oriented advanced calculus book such as courant.
i add however that i think dieudonne's approach to integration theory is very unhelpful. he assails the value of riemann's approach in that book and then later in his higher volumes gives a very abstract, complicated, and to me unintuitive approach to lebesgue integration. to be sure the elementary version in the book you cite is fine and interesting, but it unnecessarily disparages riemann integration in favor of lebesgue's, whereas the approach he gives later to lebesgue's theory is hardly one i would recommend. i.e. while i agree that lebesgue's integration theory has multiple virtues compared to riemann's, nonetheless dieudonne's choice of presentation of lebesgue's theory is not recommended (by me, admittedly not an expert) as an introduction.
to cite one example of an admirable feature of dieudonne's book, how many books you know contain a self contained proof of the jordan curve theorem, (an appendix to the chapter on complex analysis). however, my recommendation is not toe fall for his snooty prference for doing analysis over a banach space of scalars, when the complex numbers will serve just fine. to be sure he says one may assume all scalar domains are reals or complexes but he says it in a sneering way that makes one embarrassed to do so. try to ignore these rude and snotty lapses and benefit from the magnificent content of his book. it was typical in the 1960's for some reason that some outstanding mathematicians felt the neccesity to insult their readers while teaching them. to be fair, they did not mean to insult the reader but rather the outmoded members of the previous generation of textbook writers, but the reader feels it just the same.
as to bourbaki, these are books of excellent quality, written by outstanding mathematicians, but not attempting to motivate or really teach, merely to present material in a version thought to be the best available. so the material is excellent but the presentation is not at all user friendly, except for the great clarity. so what they say is very clear and correct, but why they say it that way is not mentioned. there are however potentially useful historical appendices seldom found elsewhere.
