I'm trying to find good books on the history of mathematics, dating as far back as possible.
There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to Philosophy, and there were no good recommendations on books relating specifically to Math History.
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I can vouch for the Stillwell's one : Mathematics and Its History, and here's why: unlike most math history books which are, well, focused on the history alone, Stillwell's focus on the math.
That is, it's not about the mathematicians, but about the mathematics. Each chapter deals with a math topic (be it pythagorean triples, analytic geometry, etc) and it even has exercises to help you get an idea on what the main involved ideas are.
each chapter has a supplement showcasing one or two biographies about the main mathematicians involved on the chapter's subject
But the real feature, again, it's that the book is structured focusing on the mathematics history, not the mathematicians history (hence its title)
There's also an oldie but goodie: 2 volume set Eves' Great Moments in Mathematics , part of the Dolciani Mathematical Expositions series vol1 : before 1650 vol2 : after 1650 here each chapter is devoted to help you know about a specific mathematical breakthrough and why it's important on the mathematics development
This one starts talking about the Ishango Bone, which is dated around 10 tousand years ago, given you want to go far back as possible
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As far as I know, Boyer's books (e.g. A History of Mathematics or The History of the Calculus and Its Conceptual Development) are classics. I read the first one some years ago and found it very interesting, easy to read and complete; its only flaw is that the history stops at the first half of 1900. In order to overcome this problem, I bought and read Odifreddi's La Matematica del '900 (i.e "Mathematics of the 20th century"): it is a good book, but I don't know if it has ever been translated in English.
Another good read, though a bit too specific, is Heat's History of Greek Mathematics (1921, two volumes): actually, I find this book very good, for its wonderful mix of mathematical rigor and literary style.
Todhunter's A history of the progress of calculus of variations during the nineteenth century (1861) is enjoyable, but (for obvious cronological reasons) it lacks an account of the modern Calculus of Variations, based on the direct methods; some developments can be found in Goldstine's A history of the calculus of variations from the 17th through the 19th century (1981).
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In my undergrad, we used Burton's History of Mathematics: An Introduction (the 5th edition of this). We also had as supplemental reading Dunham's excellent Journey Through Genius. This one is inexpensive and was an enjoyable read.
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Mathematics, as it is known all over the world, as a Theoretical Science, begins with the Ancient Greeks. And in that regard, the beginning you can study by reading the works of Sir Thomas S. Heath. Any of his books are great. But the most important and useful book is still his 3-volume (Dover) "Euclid's Elements, which is rich in commentary. You cannot go wrong if you begin with the careful study of Euclid as Heath presents him (in 1908 in English). Dover is a facsimile re-printer.
It rather depends on what level you are aiming at. I found Morris Kline's Mathematical Thought from Ancient to Modern Times very impressive.
If you are prepared to stop in the 19th century, I enjoyed A Short Account of the History of Mathematics by W. W. Rouse Ball when I was young, blowing the dust off an inherited copy WWRB had given my grandfather.