Weinberg's classic of cosmology still reads well, even if it is a bit dated. This is not a book for the lay person, though. The book is filled with complicated and rigorous mathematical formalisms and has an appendix that doesn't' shy away from the quantitative work that went into reconstructing the universe in the minutes that followed the Big Bang itself.The First Three Minutes
is still an enjoyable read, even after over thirty years since the original date of publication. Newer editions have an updated afterword by Weinberg that discusses "recent" developments on cosmology and astrophysics that updates the picture he painted so many decades ago, but even this afterword falls short. It fails, for example, to take into account the discovery of the accelerated expansion of the universe or the updated evolutionary models for the universe's development through the introduction of newer concepts like dark matter and dark energy. (In fact the second chapter on still suggests that the expansion of the universe should be slowing down due to gravity.) These shortcoming don't' really detract from the overall purpose of the book. The fact is, the universe is expanding and the logic that you should be able to rewind the expansion to infer a big bang and a time when the universe was hot and dense, still holds true as do most of the scientific facts and inductions Weinberg presents for the nature of the universe in its earliest configurations.
In short, the book is well organized and a popular reader with a little background in physics will find it enjoyable. Weinberg lays the foundation for his speculations and reasoning in the first and second chapter where he examines the tools and clues that he'll use to paint his picture of the early universe - namely observations of its rate of expansion and the leftover cosmic microwave background. Readers of popular science books should be familiar with these stories and concepts. The book then proceeds to examine the evolution of the universe in the first fractions of a second and first few minutes followed by a chapter on speculations for the future development of the universe. This latter chapter suffers from the same deficit of understanding as the first chapter and Weinberg's cool confidence in the future contraction of the universe and an ultimate Big Crunch scenario seems naive by today's standards and in light of the knowledge of universal acceleration.
My advice: read this book. It was a landmark for its time, but pay more attention to the middle chapters than the end or the beginning. If you're uncertain in your basic understanding of physical principles or cosmology, you might try reading A Brief History of Time: The Updated and Expanded Tenth Anniversary Edition first or even Sean Carroll's From Eternity to Here: The Quest for the Ultimate Theory of Time first. If you're a student of physics or astronomy, or even slightly more than a lay person, Weinbergs more rigorous mathematical treatment can seem like a treasure trove that further enlightens basic concepts introduced in more casual works.