Okay. So you want to learn about ancient science. Whatever should you read? Here is a short survey of what's worth the bother.
There is unfortunately still nothing comparable to George Sarton's two-volume survey of ancient science A History of Science, better known as volume 1, Ancient Science through the Golden Age of Greece (1952) and volume 2, Hellenistic Science and Culture in the Last Three Centuries B.C. (1959). Though these are out of date, they are surprisingly good for the time they were written, and they are reasonably broad in scope. If you choose to peruse these, keep in mind you will need to check more recent work as well. Also, these do not treat Roman science (so you won't find anything on Hero, Menelaus, Ptolemy, Galen, Soranus, etc.).
Next in line is the much shorter (and easier to read) two-volume set by G.E.R. Lloyd, Greek Science: Thales to Aristotle (1970) and Greek Science after Aristotle (1973). These are also out of date, but less so, and though they are much less extensive in coverage, this fact adds up to the converse advantage that they are a quick read. They also cover the Roman period. There is nothing comparable to either Sarton's or Lloyd's surveys, which is a shame since the field has advanced so much in the past thirty years that a new and improved survey of the same sort is much needed. But until one appears, these two sets are superior to any other general survey on the subject, except the following, which nevertheless are not in every respect as good:
An even shorter survey of ancient science has recently been written. This is by T.E. Rihll, called simply Greek Science (1999), though it covers the Roman period, too. This is a tiny book, as short as a single volume by Lloyd, and thus an even easier read, yet it is even broader in scope, which necessarily means it is even shallower in its treatment. Yet this book is well worth reading, as it offers decent and usually up-to-date summaries of many fields of ancient science, and thus "updates" Sarton and Lloyd in some respects.
Rihll's book has three defects. Though these are not great enough to outweigh its merits (which are brevity, accessibility, scope, and recency of scholarship), a reader should be aware of them: first, Rihll often cites sources that aren't in her bibliography (an obvious editing defect), which can be a challenge for anyone who wants to follow up what she says; second, it is way too brief to give you a really good impression of just what ancient science was like or what it achieved; and third, it is occasionally wrong (but not disastrously so). Despite all that I highly recommend it. We need something better, but there isn't anything yet that is as comprehensive.
A good supplement to all of the above is the collection of Cohen and Drabkin, A Source Book in Greek Science (1948). This is out of print, and is obsolete on many points of commentary and inclusion, but since it mostly just collects ancient passages on a very wide range of scientific interests, you will find a lot of primary source material a layman might not have easy access to otherwise (e.g. here you can read Hero's discussion of the principles of friction in machine design, a collection of hard-to-find discussions of magnetism, bits of Ptolemy's Optics, Galen's famous experiments on kidney function, ancient examples of chemical knowledge, etc.). Used editions are often too expensive, but for those who are especially keen to learn about ancient science, if you ever find a cheap used reprint, I recommend buying it. But you have to be gung ho. As it doesn't exactly make for exciting reading.
An attempt at a modernized version of this was made by Irby-Massie and Keyser, Greek Science of the Hellenistic Era: A Sourcebook (2002), but this is not quite as extensive as Cohen & Drabkin's sourcebook (whose quotations are much lengthier, for example), and often not that much better in quality of commentary or inclusion (so this new sourcebook is not all the improvement it could have been), but it does provide a good overlap and sometimes an update to Cohen & Drabkin, and might be more approachable for laymen (as it is written for beginning undergraduates).
Finally, there is Lucio Russo's The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Reborn (2003), which is a revised edition (and translation) of La rivoluzione dimenticata (1996). This is simply a must-read on ancient science. It is flawed in several respects, but there is nothing that comes at all close to it as a comprehensive, accessible, and up-to-date survey of ancient science, especially as it corrects a lot of tired old myths. Besides, it is simply a good read. I'll leave that as a sufficient recommendation, and simply catalogue what's wrong with it, which any reader should know.
First, Russo uses his survey as a base from which to launch a thoroughly untenable theory, which threads the book occasionally throughout: that Hipparchus and Archimedes were heliocentrists who had achieved a complete Newtonian dynamics that was rapidly forgotten before the Roman era. This is absurd on many levels, and easily refuted by any expert, but laymen might be vulnerable to his arguments. Just be careful to distinguish between his own unique theories (which are rarely sound), and the facts and scholarship he presents (which are usually sound), and just marvel at the ingenuity of his use of that evidence to make a case that is, ultimately, wrong.
There are a few related tendencies toward hyperbolic theorizing in the book, such as his effort to argue that the Roman engineer Hero invented motion pictures (pp. 139-41). The texts he relies on not only do not support this, but plainly refute it, and though a layman might not know that, a cautious eye will be able to tell he is proposing a novel theory that doesn't really hold up. So, too, in any other case. Like, for instance, his attempt to argue the ancients had actual telescopes (pp. 271-72 and 343-49), which is unfortunate, since Romans were experimenting with lenses, yet arguments like Russo's might cause balkers to overlook this more modest truth while rightly fleeing from his more egregious claims.
Second, because this underlying crazy theory of his requires an inexplicable and wholly implausible "mini-dark-ages" between 140 and 40 B.C. in which vast quantities of fundamental knowledge was lost (even simple biographical data on major scientists), Russo is compelled by his own logic to dismiss Roman science entirely. Although the evidence even he presents of Roman science and technology is already enough to show he is entirely wrong to slight it, the overall impact of his neglect is that too little of Roman achievements is mentioned, which is a major failing of this book.
As just one example, in this context, of how Russo can make an argument that might seem convincing to a layman but that an expert could easily refute (and there are several instances of this), he uses Ptolemy's neglect of astronomical observations in the 1st century to argue that there were none (pp. 282-83), even though we know for a fact there were a great many. Not only is Russo evidently unaware of the evidence to the contrary, but he also overlooks the fact that Ptolemy explicitly states as his method the employment of observations as far apart in time as possible in order to benefit from the greater accuracy of the intervals (correctly applying the law of averages as a means of error mitigation), hence the reason Ptolemy does not use 1st century observations is that he had no methodological need for them (though recent analyses show he probably did use them, he just had no need to cite them when establishing parameters in his published work).
Nevertheless, though Russo does badly on Roman science, and occasionally descends into maverick over-speculating about Hellenistic science, his treatment of the latter is very good, and so far has no peer (again, excluding from that remark specialized studies in specific sciences). Consequently, as long as you keep the above cautions in mind, I cannot recommend Russo's book more. It is an excellent survey that will introduce you to a good amount of ancient science.
Last but not least is a new series by Routledge that, upon completion, will be more comprehensive and up-to-date than anything else so far. So I will close by surveying this collection, because it is already the best available on ancient science for anyone really keen on learning about it. But it is still incomplete. For example, they have not yet produced volumes on mechanics or astronomy, two of the most advanced and important sciences of the time (though I believe a volume on mechanics is on the way, and surely astronomy can't be far behind). But most volumes so far are of superb scholarly quality, and are superior to anything else available (with some exceptions). Routledge generally has each book written by one of the world's leading scholars in the given field, which often makes the result thorough and definitive (though these are still only surveys and thus do not cover everything in each field, but merely present repesentative summaries of what we know).
Roger French, Ancient Natural History (1994). This is a good survey of what we call "natural history" (collecting facts about life, the earth, man, etc., so something like observational geology, zoology, botany, anthropology, etc.), though it over-emphasizes Pliny the Elder (rather than his lost sources and contemporaries), is a little too postmodern in methods and presuppositions, and needs supplementing with a sourcebook (like either of the two I discussed above).
Tamsyn Barton, Ancient Astrology (1994). The editorial guideline for this series is to produce definitive volumes on fields that were effectively recognized as sciences in antiquity. Astrology certainly rates on that measure (as would dream interpretation, for example, which was becoming 'scientific' in the Roman period, i.e. systematic and, believe it or not, highly empirical). So if you are only interested in precursors to modern (i.e. real) sciences, you can skip this one, although it is fascinating to see how astrology became 'scientific' in Roman hands and how it influenced and was influenced by other 'real' sciences. Though, like French, Barton is a bit of a postmodernist, which is occasionally annoying, if you want a good summary of ancient astrology from a cultural and historical perspective, this book is certainly definitive (as on this subject Barton has no peer).
M. Rosemary Wright, Cosmology in Antiquity (1995). This sucks. Sorry to say it, but it is an odd man out in this otherwise excellent series. Wright covers the subject poorly, far too briefly, and lopsidedly, focusing almost entirely on the presocratics (yet there are better books available on them), and spending most of what's left of her time on Plato and Aristotle. Overall I'd guess less than 10% of the book covers Hellenistic and Roman debates and developments. Strato, for example, gets no more than two or three sentences in the entire book, despite having founded a major competing school of thought in cosmology that continued to be debated into the Roman period, while Posidonius gets even less treatment than that, despite having established the most influential interpretations of Stoic cosmology in the Roman period. Epicureanism gets a mere three pages. Evidence for Roman-era debates and innovations in Galen or Sextus Empiricus or other Roman sources isn't addressed at all. This was a field in which a great deal of work has long been needed, yet all Wright did was summarize tired old observations, and not very well. Not worth reading.
John Landels, Music in Ancient Greece and Rome (1999). Landels knows his subject, and this is certainly the best summarizing history of ancient music and music theory I have ever seen, although it mainly emphasizes the actual practice and art of it, and its role in ancient culture over time, treating the science (of harmonics) a bit less than one might want. On the other hand, he covers so much of the ancient culture of music that this book is both amusing and illuminating, so I still recommend it, especially if you are into music and music theory.
Serafina Cuomo, Ancient Mathematics (2001). This is superb. Cuomo is brilliant, and her command of ancient culture and science, especially in the Roman period, is impressive. She also writes well, and keeps her subject interesting, even though one would expect math to be boring, and the history of math doubly so (though a caution: I might be biased, since I found a modern book on the history of the square root of negative one a blast). Her focus here is on mathematics, and the culture of its pursuit, and thus she only touches occasionally on the sciences per se, so if you are only really keen on ancient science you can skip this, although you will not fully understand ancient science without understanding its mathematical context (which was appreciated by, and influenced, even medical scientists like Galen).
Liba Chaia Taub, Ancient Meteorology (2003). This is surprisingly good, as one might think the subject was among the silliest in antiquity, yet Taub covers a wide range of material well and shows some of the silly, but also a lot of the genius, and the genuinely useful, in ancient theories of climate and weather. She explicitly limits herself to extant material (and thus apologizes for not discussing, through reconstruction, the lost theories of such influential luminaries as Strato and Posidonius), but no one has treated even the extant material anywhere near as well, and there is enough of it to span antiquity and thus produce a representative survey (just not a comprehensive one). She also writes better than Wright (who also ignored quite a lot of extant material), and Taub's work is more cutting edge and up-to-date. So unlike Wright, I recommend Taub.
Vivian Nutton, Ancient Medicine (2004). Perhaps the most superb contribution to date. Nutton has long been widely recognized as among the greatest historians of ancient medicine, and the scope and depth of his knowledge shows here, as does his cutting-edge awareness of recent scholarship (not a small amount of it his own). The book is long, but comprehensive, and I would even offer it as a model by which every other entry in this series should be modeled and measured, especially in terms of scope, depth, and range of content. He also writes well. No better history of ancient medicine exists.
Gavin Hardy, Ancient Botany (2007). I have not yet seen this, as it was only just released. I have it on order. I expect great things. But I'll add my two cents in the comments section below when I know more. Likewise for Andrew Barker, Ancient Acoustics (scheduled for release in 2008 but already available for pre-order). I have not seen this either, for the more obvious reason that it has not even been released yet. But it should be interesting. No one has devoted a book solely to that subject (usually acoustics is blurred together with ancient music theory and harmonics), so it should prove worth reading. I'll add my two cents on this, too, in the comments section below whenever I actually read it, as I will also do for any future entries in this Routledge series in ancient science, such as (but not only) Robert Hannah, Time in Antiquity (also scheduled for release in 2008 and available for pre-order). I would have expected this volume to be by Richard Sorabji, far and away the leading expert on the subject, but hopefully Hannah will produce a good and thorough discussion.
That covers all the books worth reading for laymen who want to learn about ancient science.
UPDATE: See now also my new book Science Education in the Early Roman Empire (2016); my chapter on ancient science in The Christian Delusion (2010); the new Encyclopedia of Ancient Natural Scientists (2009) and The Oxford Handbook of Engineering and Technology in the Classical World (2009); and the books I review in From Catapults to Cosmology.