For anyone who wants to attain the vaunted title of "being well-read," it's more about breadth than depth. (As for feeling well-read, read the postscript.)
To "feel" well-read in literature, it's all about the categories, not the books themselves. Read a few books in a few different genres, time periods, points of views. I've thrown in a few controversial books, just so you know what all of the fuss is about.