Novel and Short Story Writer's Market. Your trusty writer's bible, Writer's Market, doesn't just provide you with literary magazines to submit writing to. It also has an extensive index of writing contests. Writer's Market will tell you how and when to submit, the cash rewards, and a brief summary of what kind of work each contest is looking for. When you're getting started in the fiction contest world, start with your writer's market!
Creative Writing Contests.This wordpress blog provides nearly daily updates with poetry and fiction contests. It's an excellently run site, with detailed information about each contest, including deadlines, prize money and the site to go to for more information. Subscribe to its RSS feed to get almost daily information about contests.
Winningwriters.com This site runs several writing contests for flash fiction, regular short stories, and poetry, and has generous prizes as well.
Glimmer Train.The excellent literary magazine Glimmer Train is highly exclusive when it comes to getting published in its pages, but it runs several fiction contests throughout the year, including categories for very short fiction and a new writers category for amateurs. Consider sending your best work to the new writers contest -- you might get lucky!
That's just a start, but these links will lead you to a plethora of different contests. Start with Writer's Market for a long list of contests, get updated about the latest and newest contests from Creative Writing Contests, and try your hand at a handful of them. Don't overpay -- my rule of thumb is never to pay more than $15 for an entry fee -- and keep your fingers crossed. Good luck, writers!