I do not suggest Shakespeare, at least in the original dialect. Hell I was assigned to read Othello in college and could not understand it at all until I got ahold of a modernized translation. Modernized translation, sure, great stuff to read.
I also had a hard time reading Mark Twains books where he writes out the accent of various characters. I had when I was a kid, a bunch of classic young adult novels (Twain, Poe, Count of Monte Christo, The Bounty, etc) that were tweaked by the publishing company to make them easier to read, and they were great, very good stories that are captivating. They came in boxed sets/collections.