I second Fran. I happened to read Persuasion first . . . I had not heard of it but it was the only Jane Austen novel on the shelf at the library : ) I have since read and enjoyed her other novels and works, but Persuasion is my favorite. (Actually, my favorite novel period, by any author, with P&P a close second.)
Pride and Prejudice, Persuasion, Emma,.........Sense and Sensibility (the sensibility sister gets on my nerves after a while), Mansfield Park and Northanger Abbey aren't my cup of tea. In Persuasion, I love the longing Anne has for her Captain Wentworth. The movies just don't show that like the book does. Or I should say the 1971 movie you can see it but only after you read the book to know what to look for. I told someone else that and she didn't listen and just watched the movie and didn't like it. So I reminded her that you must read the book to understand the movie.
I would begin with Pride and Prejudice. It has a more even tone of happiness throughout and will introduce you to Ms. Austen's style. Northanger Abbey might be a good 2nd, although people do tend to consider it the least of the 6 novels. Sense and Sensibility then, followed with Emma, Persuasion, and finally Mansfield Park. I put the novels in this order because both Persuasion and Mansfield Park are slow starters and it helps if you realize there will be something which pays you back for reading the first few chapters.
I reread all of Jane Austen's 6 novels every year or two. Over the years, each has been my favorite (even Mansfield Park!). For a beginner, though, I remain convinced that Pride and Prejudice is the best place to begin. It has more humor at the beginning and a hefty serving of Miss Austen's wit with less of the discomfiting exposures and embarassments seen in the other novels. Mrs. Bennett's sillinesses excluded, of course, because what reader serious enough to attempt Austen could identify with her. Take your time with Miss Austen, she compared her prose to a tiny piece of ivory which she polished unceasingly. You will marvel at her ability to convey layers of meaning with lucid prose and will fail to find any excess which an editor's blue pencil could have improved by pruning. Each time I re-read (even after some 20 times) I uncover new marvels in her use of the English language. Bon Appetit!