I'm a native U.S. English speaker. In high school, in our English classes, we read through a number of books that were considered classics. I haven't formally taken classes since then.
I'm not sure what you mean by "classical novels," but I am a somewhat bookish person and I frequently read books written just about anytime from 1850 on, simply for pleasure. For example, last year I read "Ben-Hur, a Story of the Christ," by Lew Wallace, written in 1880; I started it out of curiosity because it was such a best-seller in its time; I'm not a religious person and I was quite surprised by how interesting and readable it was. I own "The Annotated Sherlock Holmes," an expensive three-volume set that has all of the stories and novels surrounded by notes and explanations because I like the stories a lot, and have probably read through all of them several times.
I read "Pride and Prejudice," by Jane Austen, last year, for the first time. I managed to get through it, but I don't seem to be a Jane Austen fan. I assure you though that many people absolutely do love her books, not as "classics" but simply as wonderful novels, it just seems that I am not one of them. It is probably because I am not a very "social" person myself and am not that appreciative of nuances of social attitude.