That said, if you want to start out with a very nice set of "classics" to dive into, consider getting a set of the Harvard Classics. I've got a very nice set that I picked up for a couple hundred. There are several editions. For example, on eBay: https://www.ebay.com/itm/The-Harvard-Classics-1980-Collector...
You might also consider Brittanica's Great Books of the Western World. I don't have a set of these yet, but would like to get one, at some point. Again, there are several editions, but you can pick up an older one within your price range: https://www.ebay.com/itm/Britannica-Great-Books-of-the-Weste...
If you mean _classic_ classics, as in the classical Greek and Latin canon, search eBay for "Loeb Classical Library lot." Those will definitely give you some challenging works to sink your teeth into.
Get a subscription to the Library of America and get great writing by American authors for $30.90 per volume delivered according to whatever schedule you want. The books are very nice small boxed hardcovers. I've got almost 100 now.
If you like things a little flashier and don't mind trade paperbacks, I recommend the Penguin Classics Deluxe Edition trade paperbacks. https://www.penguinrandomhouse.com/series/254/penguin-classi...
I read and collect books from the New York Review Books Classics series. It's an amazing series of reprints of mostly 20th Century literature, some that has fallen unjustly into obscurity, and a lot of it in translation.
I also pick up used sets from the Folio Society, for example Orwell's "Reportage" boxed set: https://www.ebay.com/itm/GEORGE-ORWELL-REPORTAGE-5-VOLUMES-I... which is a very nice edition that I love to hold in my hands and read. Watch prices for a while and get a sense for them, save some searches, then jump on one when an exceptional deal pops up.
Your library should reflect your mind and personality as it grows and develops over time!