Euler's two-volume Introductio in analysin infinitorum is an excellent read, from the preface all the way to Appendix on Surfaces at the end of Volume II. Here is the opening paragraph laying out the principles of the treatise, which I translated from the 1961 Russian translation:
I had noticed on many occasions that the majority of difficulties encountered by the mathematics students in the analysis of the infinite arise because having hardly digested elementary algebra, they direct their thoughts to this higher art, and consequently not only do they remain at the doorstep, but they even form perverse impressions about the kind of infinite which is used there. Although analysis of the infinite does not require perfect command of elementary algebra and all the techniques pertaining to it, yet there are many questions whose resolution is important for preparation of the students to the higher art, but which are either omitted altogether in elementary algebra or are considered cursorily. Therefore I have no doubt that the content of these books will help to abundantly compensate for the gap just mentioned. I have attempted not only to give a fuller and more precise treatment to everything that is required for the infinite analysis, but also to develop a good many questions that would allow the readers to come to grip with the idea of the infinite in an unobtrusive and natural way. Many questions that are commonly treated in the infinite analysis I have resolved here by means of the laws of elementary algebra, so that later the very essence of both this and the other method become clearer.
In spite of the modest reference to "elementary algebra", Euler's treatise gives the first modern exposition of analysis based on the notion of function, including a systematic treatment of exponential, logarithmic and trigonometric functions, and contains plethora of examples of working with infinite series. Among the highlights of the first volume: the method of generating functions is applied to the enumeration of partitions and the computational aspects of the series (including Euler's zeta function) are given a superb treatment not reached in the contemporary curriculum until a good course in numerical methods, if then. The second volume is devoted to coordinate geometry of curves and surfaces and lays down the foundations of differential geometry.