When I started delving in to The Elements I had the same issue. Euclid's propositions are somewhat unusual especially due to the translations. I actually did a whole blog post on this that was geared towards new students trying to learn and understand geometry. I go over the propositions in it if you are interested: http://mathhelpblog.com/geometry-for-dummies-how-to-do-geometry-and-understand-it/. Perhaps this could serve as an example on how to show it it more modern terms.
It's a challenging read, but I figure since it was the go-to math text for over 2000 years, that it still has its relevance. Euclid is a personal hero of mine. I know its been a while since you wrote this post, but I am interested in what you came up with.
Victor Katz is renowned for his writing and research in the History of Mathematics. I read an earlier edition of the text History of Mathematics while taking an undergraduate course in the History of Math during a Spring term (it was the required text for the class.) It is an excellent book. We couldn't cover the entire text over one semester, so I persisted in reading it to completion over the summer which followed.
You can go as far back as you'd like (it goes very far back in history!), or pick up where your interest is piqued.
If you can't take the course, for credit, or as an "auditor", I'd recommend this book for your library. It is a good complement to "doing" hard-core math. That's not to say that it's necessarily "easy", because it invites you to engage in mathematics using only the tools available at a given point in history and in a given culture. At any rate, I found the text to be very engaging, it helped me appreciate the field of mathematics more than I ever thought I could, and it has served me well as a reference, too.
I just noticed that there is a "brief" version of Katz's History of Mathematics which might not be as overwhelming, and likely highlights the best and the biggest breakthroughs in mathematics, as they developed over time.
You might also want to peruse the following list: Resources: History of Mathematics, to find some helpful recommendations.
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It depends on what you mean by error. The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof. Further Euclids definitions, although nice sounding, are never used. We now know that there must be undefined terms in an axiomatic system. Finally Euclid did not treat the issue of order. Hilbert's axioms are a completion of Euclid in that he gives all undefined terms and all axioms necessary for geometry. Ironically, Euclid was right about parallels, the one thing for which he was criticised for centuries.