Speaking of The Iliad and The Odyssey, are there any particularly good English translations of it that I could read? Is Penguin Classics good enough? Are there better ones?
That's why I'm always hesitant to read non-natively English texts from Project Gutenberg. It's often a translation from the 1800s, or something that is, as you say, olde-timey. That's fine for English-native works, but it grinds me a bit with translated ones.
When it comes to Shakespeare I just enjoy the originals as is. I read a lot of the plays and found the stories extremely enjoyable. I haven't seen many productions of the plays, however.
That said, I also appreciate movies like 10 Things I Hate About You as a rendition of Taming of the Shrew, or that Richard III where Ian McKellan is a Hitler-esque fascist.
I think younger students maybe should read the modernized versions (with some explanations for anything quirky that got lost), but the older ones should be trying to go through the original English. There's just so much to absorb on even a first run through if Shakespeare's plays (I'm always saddened when I try to reference stuff and almost none of my friends read or remember any Shakespeare), just the historical connections alone are really great/important. I've wanted to suggest watching HBOs Rome so many times to people but they never went through the Shakespeare plays and it's a much harder sell
Part of that is because, with a few exceptions, the plays were already anachronistic and it's not like early modern England was a place that was big on the historical accuracy thing...