I like benefitstaxcredithelp's list, because those are all classics that are also quite readable in modern terms. More generally (and speaking as one with graduate degrees in literature, who genuinely loves sitting cosily reading an 800-page 18th century novel):
read what grabs you.
Classic can mean a lot of things, from a massive volume by Dostoyevsky that might not even have been well translated, or Moby-Dick (which I personally find hilarious but is so slow on action that the whale in question doesn't even show up till page 600), to, say Kazuo Ishiguro's Never Let Me Go or The Remains of the Day. Both of those were written in our lifetime in modern English but led to his Nobel Prize in Literature (which, honestly, I don't think he got for, say, The Unconsoled! though technically the Nobel is given for a body of work and not just one or two specific books...)
I would argue that the first and best Jack Reacher novel by Lee Child, "One Shot," is a classic in its thriller genre, though it probably won't show up on an A-level curriculum soon. This is why I thought benefitstaxcredithelp's list is good, it focusses on relatively short books written in modern English that are definitely still generally agreed to be good even by the university nerds.
If you're feeling intimidated,
--start with shorter classics, written in the 20th or 21st centuries;
--look for ones that have also been made into films and go ahead and watch the film first, then read the book to see how it's different (especially with, say, Shakespeare plays, which were MEANT to be seen performed, not read),
--and consider reading the plot on Wikipedia or the like first, so that you can relax and enjoy watching how the story unfolds instead of impatiently waiting for what happens.
And use your library if you have a decent one (try consulting the librarian, he or she will be thrilled) or your charity bookshop. That way, you won't feel guilty trying and abandoning a few books that just don't work for you.
Remember, you are not studying for an A-level or a Ph.D. (are you?) -- reading is meant to be fun. Forcing yourself through books you don't "get" or don't enjoy will actually put you off. There are plenty of classics you probably will love. If and when you get around to trying the big long 19th-century novels like Bleak House, which take a while to get moving on the actual plot, it will help if you've kind of worked your brain muscles up to it, like prepping for a marathon with a few solitary jogs in the park. And if you still can't face Bleak House, no shame. Plenty of people just don't like any Dickens and the most snobbish thing of all in Ph.D. programs is to say so shamelessly at cocktail parties...
Don't start with too rigid a list or any deadline. This isn't a job, it's supposed to be fun. Once you've read one book you really liked, whether Jane Austen or anyone else, you can try another of their books and see it it's as good or better. Plus you might poke around and discover that people were just as condescending about Jane Austen in her day as they are now about, say, Sophie Kinsella they called it "chick lit" and not serious. It wasn't in Latin, didn't even quote any classical Latin or Greek authors, and was about women, not men, about marriage, not politics or God.
Really really. Reading is not inherently more virtuous than watching a film or being with people you love or cooking dinner. (Kind of less, actually. Isn't reading kind of... solitary and selfish?)
As I say, I got graduate degrees (in the plural!) in literature. I still give up all the time on books that aren't working for me. Sometimes I go back to them years later and for whatever reason, my mood of the day or how much I've changed since or even (whisper it) a copy with whiter paper or better typeface, I gobble them up. But life is long and no one has time to read everything.