To me, being well-read does not neccessarily mean having read all the classics which most people have not. It does mean that one should recognize many of the classic themes and storylines and characters.
I would encourage you to keep your eyes and ears open for books, authors, poems, poets music, composers, even movies and plays that are referenced by others. Then find them and read (or otherwise expose yourself to them). If you like one work by an author (or whoever) read more. If you don’t move on.
For example, I picked up this book Suspense and Sensibility A Mr. and Mrs. Darcy Mystery by Carrie Bebris the other day. It’s a mostly enjoyable lighthearted mystery with a touch of the paranormal set in Jane Austen’s England featuring the former Elizabeth Bennett and her husband Darcy.
If I were trying to become well-read, I might take this as a clue to read Jane Austen’s Pride and Predjudice. (As it happens, I have read it and enjoyed it, more than once).
The urge to read Austen would be encouraged by noting the reference on the book jacked to a series of mysteries featuring Miss Jane Austen herself as detective, and the fact that I’ve somewhere seen a more traditional sequel to Pride and Predjudice and I know I’ve read a science fiction short story or novella featuring the Bennett sisters.
And that is not all the places that that particular novel and its characters have been reused.
Likewise, if you’ve never read any Sherlock Holmes stories, you should do so. Shakespeare’s Hamlet (as well as several other of his plays) have at least in their broadest senses become threads in the tapestry of pop culture. I’ve read several science fiction books which quote from Gilbert and Sullivan.
I would encourage you to look for those kinds of pervasive elements and look for their sources, and then read the sources. Read some mythology. Read some Ovid and Balzac and Homer.
Think about authors you read in school, read some other works of theirs. Don’t set out to read everything in a particular niche (unless that niche really appeals to you), but set out to read a smattering here and there which exposes you to a broad range of time periods, locations, styles, etc.
And don’t stop reading what you enjoy just so you can call yourself well read.