The best thing to do is go to a smaller bookstore, not a chain if you can find one in your area, and ask the employees there what books they would recommend. Librarians will also help with this. In fact, libraries have all kinds of lists of award winning books! And they're FREE!
Once you have some books to look through, find one that has a topic that really interests you. You don't state what your interests are, but "better" books doesn't have to mean Dickens. Now I like Dickens, but he isn't for everyone.
You don't have to turn to classic lit. to enhance your reading, after all. There are all the modernists and post-modernists. People like Hemingway, Fitzgerald, Faulkner....or Ray Bradbury and Ursula K. le Guin! Flannery O'Conner wrote as part of the whole Southern Lit movement back in the 1940s thru 1960s. In fact, there's all sorts of so-called regional writers out there. People like Willa Cather and others.
If you like really current fiction you could read some of the so-called ethinic writers such as Toni Morrison, Alice Walker, Sandra Cisneros, Leslie Marmon Silko.
Then there's also poetry. Don't underestimate the power of poetry to help you expand your language, your thinking, and some people even find it soul-inspiring and uplifting (depends on what you read...some if REALLY depressing - LOL)! So you could check out Frost, Adrienne Rich, Joy Harjo...the list goes on and on.
Then there are current retellings of classics! Like Seamus Heany's Beowolf.
You could also pick up Wicked, or The Ugly Stepsister by Gregory Maguire. Lamb by Christopher Moore. The Drowning Season by Alice Hoffman. Housekeeping by
Mariann Robinson.
I guess a good on line place to start, maybe before the library trip, might be with Amazon.com and their LISTMANIA feature. Those are people like you and mean who have read books and made lists by topics of interest.
Just find one book to start and read it one chapter at a time and each new chapter will lead you further into the story, and further into the area of "better fiction" that you want....and painlessly!
Good luck!