In the Preface of the first German Edition of the book Problems and Theorems in Analysis by George Pólya and Gábor Szegő, one can read [emphasis mine] :
The chief aim of this book, which we trust is not unrealistic, is to accustom advanced students of mathematics, through systematically arranged problems in some important fields of analysis, to the ways and means of independent thought and research. It is intended to serve the need for individual active study on the part of both the student and the teacher. The book may be used by the student to extend his own reading or lecture material, or he may work quite independently through selected portions of the book in detail. The instructor may use it as an aid in organizing tutorials or seminars.
This book is no mere collection of problems. Its most important feature is the systematic arrangement of the material which aims to stimulate the reader to independent work and to suggest to him useful lines of thought. [...]
The origin of the material is highly varied. We have made selections from the classical body of knowledge of mathematics and also from treatises of more recent date. We collected problems which had in part already been published in various periodicals and in part communicated to us verbally by their authors. We have adapted the material to our purpose, completed, reformulated and substantially expanded it. In addition we have published here for the first time in the form of problems a number of our own original results. We thus hope to be able to offer something new even to the expert.
To put it briefly, I think that the description of the key features that distinguish this book which is given in the preface is perfectly appropriate. With respect to these aspects, I regard this book as a sort of introduction to the methods, the ideas, and the results of mathematical research.
It is actually the first book of this kind which I have ever come across, and I've really enjoyed going through it. So, the natural question that I pose is:
$\color{#c00}{\text{Question:}}$
Are there other similar books (in analysis or also different topics) that are in the spirit of Pólya and Szegő's work? That is, are there other books that are actually introductions to mathematical research in the sense which is implied by the emphasized parts of the passage quoted above?
Note: Your question is really a challenge, cause the book you're pointing to as reference is a first-class evergreen of highest rank.
So, I was thinking: Which books give me a similar feeling when I am going through them as the classic by Pólya and Szegő and which of them could also play in the same leaque? One other criteria was, that they should provide a reasonable thorough survey through a part of mathematics.