His reputation preceding him, with a pretty good idea of what the main arguments of the book are, it is a daunting task to read Popper in the original. But it soon becomes clear that there is enough value, ingenuity and originality in "The Logic" to merit reading the original book in lieu of, or in addition to, later clarifications, commentaries and critiques (including some by Popper himself).
First, a warning. The logical complexity and mathematical sophistication of his arguments, combined with the author's engagement with the partially outmoded terminology of the positivists and probability theorists, means that some sections of the book will almost certainly appear, to all but the most savant of savants, varying degrees of impenetrable, inscrutable and unreadable. The dense appendices, full of logical and mathematical proofs, can happily be skipped. It is also clear that some parts of the main text of the book have become severely outdated, especially the sections on quantum mechanics (cf. the Heisenberg-Bohr interpretation and its implications). The low point of the book, from a readability perspective, namely the 70 dense pages devoted to probability theory, especially that of J.M. Keynes, and refuting its implications for science, seem to be of little interest to contemporary debates, and only marginally relevant to Popper's main arguments.
So I would not fuss about reading every chapter, footnote or appendix. The main gist of the book can be expressed in non-technical terms. And Popper himself does a wonderful job explaining his position in the few opening chapters. So it might even be enough to read through chapters I-VI.
At his best, Popper is a very lucid philosopher of science, whose revolutionary doctrine, that scientific doctrines can never be "verified" or deemed "true," but only temporarily more or less corroborated, is carefully argued for, using a combination of natural language and technical terminology (which is sometimes helpful, although never absolutely necessary, or so it seems to me). The fact that he spends considerable time attacking his contemporaries - the inductionists, the positivists, the Vienna Circle, the pragmatists, the conventionalists, etc. - means that he occasionally gets bogged down in technical minutiae, but the main thread is luckily never lost.
Disentangling the the fact that he revolutionized the history of the philosophy of science from the particular arguments advanced in the book is a difficult task. As Popper himself always emphasized, there are no isolated observations that are "theory-free", but we are always carrying with us some "metaphysical" assumptions (strictly speaking untestable, unscientific and unempirical). Thus, before reading the book, the reader will almost certainly have been introduced to Popperian methodology via some circuitous route, e.g. a commentary, a philosophy course, or a later critique. These form a king of "background halo" through which the reading experience will be refracted.
Some of the "baggage" that I personally brought to the reading experience - in additional to a deep respect for Popper's critique of inductive science - was the importance and value of some of the criticism that has been targeted against him. This is no time to go through the theories of Kuhn or Feyerabend, for example, but I'm sure the reader will find the time to go read those authors. The fact that Popper has influenced subsequent scholarship is vital to understanding him. It WILL colour the reading experience, but not in the sense of "tainting" the pure original picture, but "motivating" the reading experience with a combination of prejudiced respect and curiosity.
However, I wish to offer one major criticism that I have with Popper's wonderfully stringent theory: I worry that Popper offers too much of an "idealized" version of science. He proceeds logically to devise an ideal system that bears little resemblance to actual science. More concretely, Popper's "system" (if we can call it that) contains a realist component and an idealist component, and these two aspects do not mix easily. In fact, they are often in dire conflict with each other.
The realist component, which I admire, is embedded in his notion of how hypotheses and theories come about. The idealist component, which I see as the problem, is embedded in his notion of how hypotheses and theories are supposed to be put into the test (ideally of the potentially falsifying "crucial experiment"). Overall, Popper's view of science describes an ideal scientific practice, whereby theories - motivated by WHATEVER source, be it rational belief or irrational flight of fancy - are devised, and crafted, in such a way that they meet the maximum criteria of universality, simplicity and testability (i.e. falsifiability). This is certainly a difficult task, but Popper insists on it. In this view, doing anything else - i.e. devising "ad hoc" explanations to explain weakness, or devising metaphysical tautologies to explain strengths - is frowned upon, and the practice of science is hopefully purified of such bad methodologies. But we have a problem, since there is no guarantee that the scientists, who are psychologically very much invested and often in love with their own theories, will, absent strong institutional incentives, be motivated to design their theories in such a manner. This assumes too rosy a picture of humanity. In his "Open Society and its Enemies", Popper has accused social scientists of being naively optimistic. But in "the Logic", Popper seems to fall for the same fallacious view of human motivation and capabilities. Even if Popper acknowledges his logical focus, he SEEMS to be implying that devising such a system of science, where fallibility and falsifiability are crowned king and queen, is theoretically possible.
But human beings have a psychology that is fighting hard against such devises. Scientists, like other humans, will carry their irrational and self-serving practices, which motivate their quest for knowledge, and their desire to build metaphysical systems, all the way to the experimental stage; and only a strong community of peer pressure and external criticism can build sufficient safeguards against building theories that are impervious to refutation, or, which comes to the same thing, of being reluctant to accept studies that threaten one's cherished theories. Whether such strong institutional studies can be built is an empirical question; even if Popper provides a good way for an ideal scientific community to be structured, it is dubious whether it ever will be. A more reasonable assumption is to assume that science will, hopefully, come to approximate a system of perfect competition, where systems (theories, hypotheses) can be freely tested. Even if most scientists will be reluctant to set their theories to the test, peer pressure can, ideally, force it on them. This seems to me to be the only way to rescue Popper's methodology from the pitfalls of human self-deception.
And indeed, Popper knew this: he references "natural selection" and "freedom" as prerequisites of science, since they foster competition that can lead to the rejection of bad habits and theories. Indeed, Popper's insistence on falsifiability ONLY makes sense if embedded into this context. But such a context does not preclude pragmatic, instrumental or positivistic theories, either, which seems to make Popper's insistence on a particular methodological commitment metaphysical (which he readily admits); and, more dangerously, superfluous, if RESULTS are what matter. (And results can be obtained using probabilistic logic, positivistic induction, or any other methodology.)
Even if Popper's original formulation of scientific practice can be criticized, and has been with justification, this is partly missing the point. The value of Popper's insight is that even though it contains many outdated and outmoded paragraphs, it opened up a whole new methodology. Very few thinkers have been able to set up a new paradigm that challenges the very foundations of our commonly held beliefs and practices, but Popper certainly did, and he did so with rigorous logic.
Great thinkers are often ones who use reason AGAINST reason, use logic AGAINST logic, i.e. to show the limits of reason and logic, and this is exactly the greatest contribution of Popper's critical method. He does not leave us with debilitating skepticism, but with exhilarating awareness of the imperfections of our knowledge, the constantly challenging nature of reality, the critical nature of science, and the "irrational rationality" of our own drives, beliefs, faith and curiosity.