I've been a member of Bookmooch for about 8 years now, and while it's not always gratifying in the short term, it's been amazing over the long haul. Eventually, books I'm interested in come around, and in the meantime, I can send excess books to good homes that will care for them (and then perhaps pass them onto another loving home).
Recently, I've acquired Beggars and Choosers by Nancy Kress and Kitchen Confidential by Anthony Bourdain, which were on my wish list for years, so it's not the fastest method but it gets results!
2. Little Free Libraries
I'd vaguely heard of Little Free Libraries when I discovered one in my neighborhood. I always enjoy checking out the selection, and it changes often enough that every once in a while, I see a book of interest. I've also added a few books to the collection.
Recently, I've acquired Paper Towns by John Green and Girl in Translation by Jean Kwok. There's no guarantee that a book you want will magically appear, but browsing is part of the fun!
3. The Library
And then, there's the actual public library! Although I occasionally have some success getting specific books I want, I've found that most of the fun of the library is browsing or looking for popular books from years ago. If you really want a current bestseller, there's probably a long waiting list or if you need a specific book now, it's not often the best way to go, but to walk in a roomful of free books is itself a pleasure.
Recently, I've been checking out audiobooks including Anne Tyler's The Accidental Tourist. It's a good deal since audiobooks are usually expensive and, unlike physical books, I don't find I like to re-read them.