It has been my general desire for a few years to acquire the basics in other European languages for the purpose of reading some of the classics in their original language, in a similar vein to this topic. I never pursued a whole lot, and so my knowledge of exactly what those classics might be for a particular language never developed very far. In anticipation for a trip to Lyon this summer I have begun to learn a little French, and would be very interested in reading some of the more palatable (in the sense of a reader who is fairly naive to the language) French texts. My first instincts would be Cauchy and Lebesgue, seeing as I am more analytically inclined, but I have no idea where to start or which of their works are readily available.

Since you mention Lebesgue, I would recommend the following two classics, which build on his lectures at the Collège de France : Leçons sur les séries trigonométriques Another suggestions : Topologie générale by Bourbaki, and Théorie des distrbutions by Laurent Schwartz. 


The canonical excellent French author is Serre (his books are also quite easy to find)  Cours d'Arithmetique has some analytic content, if you like that sort of thing, as you say you do... 


I suggest that you have a look at Bourbaki's talks here as they range quite a few topics, are generally short enough, are often in french, and are regularly from masters. Of course, you'll find other interesting collections on the same website. 


Roger Godement. His courses (in analysis, differential geometry, algebra, etc.) are magnificient. 


If you're looking for French classics, I would recommend Darboux's Théorie générale des surfaces There is a lot of analysis there. In fact the text is mostly about the interplay between differential equations and differential geometry. Goursat's Leçons d'analyse are also quite nice. I read somewhere that Bourbaki started as a rejection to this text, but that only makes it more interesting. I also like Appell and Goursat's Théorie des fonctions algébriques et leurs intégrales. Appell's books on mechanics are really nice as well. 


Actually Borel wrote a series of very nice little books, around 1900's. One of them is called "Sur les series de Taylor a coefficient positive". It has some very nice theorem, many of which are forgotten at this day; it reads like a beautifully written paper that just came out. Also, Paul Levy. He has an exceedingly beautiful writting style. His 7 volume collected works should be available in a math library. 


Cauchy's Cours d'Analyse is beautifully written, and good for the "analytically inclined". His treatment of infinitesimals is very interesting, and it contains the famous "mistaken proof" that a limit of continuous functions is continuous. There's a CUP reprint, and it is online here 


Fourier's Théorie analytique de la chaleur is available online here. As an advantage, the English version The Analytical Theory of Heat is available here.



For the really analytically inclined :), while I have not finished reading it myself, I have been told by multiple of my French colleagues that Leray's original paper on NavierStokes has interesting mathematics and quite penetrable language. 


I'm not fit to properly judge the quality of the language, but I've always found Dixmier's papers very lucid (although mathematically demanding for this Bear Of Little Brain). Plus, one gets to see some of the theory of Von Neumann algebras at an interesting time. Looking on NUMDAM ought to yield several papers, including IIRC the paper on $C^k$ functional calculus for selfadjoins elements in $L^1$ of a nilpotent group. 


For operator theory, Dixmier seems to be a good option. 

