I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely free, translation and guide.)
We are up through Proposition 17 now, and although this is very instructional for both the students and myself, I sense that before too long the novelty will wear off and it will be good to return to a modern treatment. (I will, though, want to say a little bit about his treatment of parallel lines and elliptic and hyperbolic geometry.)
That said, there are some gems. In Book 2, Euclid constructs square roots, and in Book 4 he describes how to draw a regular pentagon (which is surely not obvious). And, the constructions in the first few books are all very interesting. (Of course there is lots of fascinating number theory too, but my course is on geometry.)
What other particularly fascinating tidbits do the Elements contain, which it may be easy to overlook?