Short review
As someone with little background in the philosophy of science, I found this book very readable and informative of the broad ideas philosophers have had about what science is and how it should work. If you are pursuing a career in science, it seems like a no-brainer to have an overview of the scientific enterprise at large, and of some of the major epistemic challenges faced in science, e.g. relating to induction, evidence, and belief. These are topics that this book approaches with clear language and plenty of examples. I really appreciate that Godfrey-Smith avoids sesquipedalianism (unlike myself) and dense language; generally, what he means seems quite clear.
Longer review
My single biggest takeaway from this book gets put forward already in chapter 1, namely Godfrey-Smith's discussion of what he regards as three defining features of science: empiricism, mathematics and its social organisation. Empiricism and science's social organisation receive much more attention across the subsequent chapters of the book, whereas the mathematical and quantitative aspects of science do not. I personally found this to be a good thing, as I already hear a lot about the importance of data and mathematical modelling by virtue of being immersed in the 21st-century era of AI and computing. In any case, Godfrey-Smith portrays science as an open and collectively driven enterprise wherein we try to produce detailed (and when possible, mathematical) descriptions of our world in a way that is sensitive to experience: in particular, observations should not just be "resources" for scientific arguments, they must, at least on occasion, effectively settle debates, and thereby serve as real constraints on our beliefs (if I observe $X$, I will believe or report $Y$). This is how we can bring the world to "bear on our theories" by bringing our beliefs "into contact" with reality. Indeed, in section 14.1, Godfrey-Smith notes (p. 328):
Maybe our shared biology is enough to make us all fairly empirical when we are trying to get food to eat and work out how to get home. But this does not apply to attempts to develop and justify theories about our overall place in the universe. The scientific strategy is to construct ideas ... in such a way that exposure to experience is sought even in the case of the most general and ambitious hypotheses about the world ... one looks to experience as a constraint on hypotheses.
All this talk is, of course, much easier said than done, and a large portion of the book looks into deep issues regarding testing (e.g. the problem of holism: our beliefs and predictions are not isolated claims, and all of our observations still need to be interpreted. For example, given an unanticipated observation, who is to say whether some theory is wrong, or whether a background assumption about the set-up in which the observation was generated did not hold), induction and generalisation (e.g. Goodman's new riddle of induction: when we seek to generalise, the data always seems to underdetermine what remains viable, and in particular, certain absurd theories always seem to remain possible), the potential incommensurability of different theories (which, especially if they are from different research traditions, might have such seemingly large differences in their standards for evidence, and in their theoretical concepts and entities, that it may seem impossible to adjudicate between different theories by "observation". In particular, both theories may interpret the same observation very differently, with both claiming to account for it, making it seem as if observation itself cannot play the role of a neutral arbiter), and other issues related to the challenges of bringing experience "to bear" on theory.
Overall, the book provides a broad overview, as Godfrey-Smith delves into many topics. We get to cover both the history of science and of the philosophy of science, and are introduced to key figures and groups such as the Logical Positivists, Popper, Quine, Kuhn, Lakatos, Feyerabend, the Strong Program, Science Studies and more, as well as influential philosophical positions such as empiricism, relativism, and naturalism. In the latter half of the book, Godfrey-Smith also delves into interrelated topics such as scientific realism (do our best scientific theories tell us what the world is really like), explanation (in terms of laws, causation or unification), and truth (in terms of correspondence, coherence and pragmatist views). Throughout the book, we learn about key concepts and terms, such as the analytic-synthetic distinction, hypothetico-deductivism, inference to the best explanation, falsification and the demarcation problem, scientific paradigms, normal science and revolutions, research programmes and traditions, natural kinds, causation, Bayesianism and Subjectivism, deflationism about truth and Occam's razor (and more). However, while we get a broad overview of many ideas, we naturally sacrifice depth for many topics.
For example, the possibility of progress in science is a recurrent question that gets plenty of treatment, as highlighted in the discussion of Kuhn (chapter 4) and scientific realism (chapter 10), but a related and perhaps even more basic question is to ask under what conditions (or rather, assumptions) scientific knowledge can be obtained at all. To be fair, this matter is extensively discussed in terms of social structures, norms and incentives (e.g. via a sociological angle under Merton in chapter 7 or Hull in chapter 9), and all of the discussion of evidence and confirmation is arguably relevant to this question. Still, for a book on the philosophy of science, one may think that there isn't that much of a philosophical analysis of what assumptions we and the world may have to satisfy in order for science to work and produce "truthful" knowledge about the world. Do we need to assume some notion of stability or invariance, e.g. that under identical conditions, a phenomenon will always evolve identically (in some sense), and what about intelligibility? Do we need to make certain assumptions about the world vis-à-vis our own minds, and on the kinds of contents we can think of? Or can we learn rigorously about the world even if our access to it is limited? Godfrey-Smith, being more aligned with naturalism, as opposed to foundationalism, would presumably have a take on all these questions that draws upon our scientific understanding of the world itself. However, diving into too many such details would go beyond the scope of "an introduction to the philosophy of science", and so understandably, there are such topics Godfrey-Smith does not elaborate on.