I will join with anatoly in disagreeing. The Elements are a beautiful piece of literature and I recommend reading them for a historical understanding. (I have not read them through by any means, but I have read segments which are magnificent).
As for understanding mathematics and its techniques, there are now much better methods. Our understanding of mathematics and methods of learning (not to mention our simple culture in general) have changed substantially since then, the most obvious point being the development of non-Euclidean geometry.
To add to the list of books already mentioned consider Numeracy. It was influential for me when I was an undergraduate and touches on numerous topics. Also, if you want to start at the beginniner, there is a very good book called Chapter Zero that starts with nothing and is easily accessible to anyone willing to study it.
You say that "our understanding of mathematics and methods of learning have changed substantially since then".
While I of course agree that we now have a lot more mathematics, including non-Euclidean geometry, I don't see what thay has to do with the value of Euclid and the other ancient mathematicians. I'm not recommending Euclid and Appolonius as a complete mathematical education, but a good place to start.
What are the methods of learning that we now have that invalidates Euclid and the others from being valuable teachers of mathematics?
Invalidates? Absolutely none. Good mathematics is eternal and the Elements has some beautiful mathematics.
What I meant to say is that in the fullness of time they have been expanded on and surpassed. You can still learn good and valid mathematics from Euclid, but you can learn it better form more modern sources.
As one example amoung many, good modern geometry books will let the reader know about the existence of non-Euclidean geometry and its differences, even if the focus is on traditional Euclidean geometry.
Also, modern books will use modern notation. Notation of course does not affect the ideas which are what is important, but someone trying to learn math today will need to learn the notation so that they can read other more advanced books that will assume the reader knows that stuff. Not to mention, notation can make things easier or harder (try doing multiplication in Roman Numerals).
Finally, many things that were proven tediously before have easier proofs developped over time, often with the ability to take recourse in Algebra. For the same reason, many open questions in Euclid's time now have answers.
However, notation actually does affect the ideas, and deeply. Euclid did not consider 1 a number, for perfectly sound reasons. Multiplication is a very different animal in Euclid's math, and his proportions are not just funny-looking fractions.
Overall, you are right. If someone is looking to get a broad survey of mathematics, Euclid is not a good source. But to start from scratch and learn rigorous math-by-proof, he has no equal.
As for understanding mathematics and its techniques, there are now much better methods. Our understanding of mathematics and methods of learning (not to mention our simple culture in general) have changed substantially since then, the most obvious point being the development of non-Euclidean geometry.
To add to the list of books already mentioned consider Numeracy. It was influential for me when I was an undergraduate and touches on numerous topics. Also, if you want to start at the beginniner, there is a very good book called Chapter Zero that starts with nothing and is easily accessible to anyone willing to study it.