Jump right into the action. Your hero could start the story running from a bull or cursing as he hits the floor after falling out of his chair. Instant conflict using active verbs will paint a scene faster than a lot of flowery, descriptive prose. If the very first sentence is rewarding, readers will naturally want to know what happens next.
Provide a distinct narrative voice by giving the narrator vivid characteristics: Holden Caulfield of The Catcher in the Rye spits his jaded view at the audience in the form of casual cursing. A few of the novels of Kurt Vonnegut---Breakfast of Champions and Cat's Cradle---begin with an extremely conversational tone. The narrator---seemingly an extension of the writer himself---acts as an experienced and authoritative storyteller, and therefore, you believe him to be.
Raise a question to draw reader interest. Make a bold, discerning statement pertaining to the story. Make the audience yearn for some sort of solution or outcome right from the beginning. Some stories have even given away the conclusion at the beginning (the movie American Beauty). Putting the reader in a state of curious confusion will give immediate direction (and possibly a payoff later when they realize the foreshadowing). Don't give up all the answers---show that the story is leading somewhere.
What makes your story unique? Does it take place on Mars (describe a red planet sunset) or have a deaf protagonist ("The crowd at the bus stop began to yell and I couldn't tell why")? You may admire those classic writers---and reading them certainly offers a lot in the way of inspiration---but strive from the start to make your story stand alone.