Consider starting with a quote or bit of dialogue. Stories that dive right into conversations between characters can grab readers' attention and pull them in immediately. Clearly identify each speaker so the reader doesn't get lost or confused. Don't, however, let the dialogue drag on too long without incorporating physical action or other description. You'll lose readers quickly if your dialogue exists in a vacuum for several paragraphs or pages.
Keep it simple. Many beginning authors overwrite their openings---and their fiction in general---thinking impressive, flowery words or numerous adjectives make their writing better or more literary. The American Book Review's top one hundred opening lines all incorporate simple, clean language to invoke an emotion in the reader. Craft straightforward sentences that shed light on your characters and your story's themes. Being a feeling writer is more crucial to good fiction than trying to be a genius writer.
Cut to the chase. "Give your readers as much information as possible as soon as possible," advised novelist and short-story writer Kurt Vonnegut. In the first few pages, readers should have a strong understanding of your protagonist. They should know her name, approximately how old she is, her distinctive traits, her passions and her primary obstacle(s) to obtaining what she wants. This all can be accomplished without beating readers over the head with a list of facts. Patricia Highsmith, author of mystery classics "Strangers on a Train" and "The Talented Mr. Ripley," lets readers into the minds and lives of her characters subtly and right from the start. "She glanced across the room and tried to think of something else," Highsmith writes on page two of the novel "The Price of Salt." "Of the beautiful black and red Norwegian sweater she had seen at Saks and might buy Richard for Christmas, if she couldn't find a better-looking wallet than the ones she'd seen for twenty dollars." In two sentences, Highsmith tells readers a great deal about her protagonist's class, religion, tastes and priorities---and without any of it coming off as a laundry list.
Edit. After you've written your story, review the beginning and cut all extraneous words, especially unnecessary adjectives and adverbs, superfluous insights into human nature or the state of the world, and vague or overused words such as "good," "bad," "beautiful," "wonderful" or "pretty." Delete or replace clichés such as "It was hotter than hell," "She let the cat out of the bag" or "He had a lot of heart and soul." If you're unsure whether a phrase is played out, go to clichesite.com, the Internet's largest collection of English-language clichés and their meanings. The strongest writing uses as few words as possible and as specific words as possible to convey its messages to the reader.