Visit the author's website. Look for upcoming signings, lectures and appearances. Autographing books is often a component of these events, especially if the author/illustrator has a new book out. Also try contacting the author/illustrator directly by sending a professional email--many of them would be happy to exchange a book via postal mail for a signing.
Contact the book's publisher. Their business is to sell books, and if you'll buy a copy, they will communicate ways to get one autographed. If the author is deceased, the price is directly correlated to the book's reputation and the availability of an edition , and it will be more difficult to find an autographed copy.
Check out your local bookstore. Many booksellers have shelves of new, autographed books; look for the corresponding sticker on the book's cover. You have a better chance of scoring a signed book if the author has recently made an appearance at that particular bookstore. If you are looking for a specific autographed book, ask a store employee.
Browse bookstores that specialize in rare antiques and collectibles. In addition to having the fun of discovering old editions of your favorites, you can often get a good deal on an autographed copy.
Attend national writing conferences and book events, festivals and meetings. Major literary events attract big-name authors, publicists, publishers, librarians and booksellers. These meetings offer multiple opportunities to meet authors, buy books and obtain an autograph.
Peruse used book sales held at libraries. Many of these books have been donated-- by patrons, stores and authors-- and it's not unusual to discover an autograph inside the front cover. Although a personalized statement is not as valuable as a solo autograph, it still adds a unique detail to a used book.
Read book blogs, buy from online booksellers and enter book contests. Many literary blogs have active book giveaways, often of autographed copies. Check out Book Club Girl, Book Slut and Autographed by Author. Search on eBay, Biblio and Amazon for listings of autographed books (especially collectible antiques). Love to read? Become a book reviewer yourself. Often, authors and publicists will send reviewers just-released, autographed books to read (for free!).