My first jazz teacher, Bob Keller, just published The Joy of Fakebooks, a little guide on navigating the various features of all the different jazz fake books available today. (I wrote about Bob Keller's Jazz Page a couple years ago. He maintains a popular, growing list of organized links to helpful jazz resources.)
One of the most important lessons I've learned since studying jazz was to learn tunes. Building repertoire is great for knowledge and marketability, but all the most important musical lessons come from real music. Studying tunes is the best way to study theory. Since dedicating myself to studying tunes, I've accumulated several fake books, including Hal Leonard's Real Book Series and Chuck Sher's New Real Book Series. That's a lot of paper between those six books. There are digital options, but until you have a computer screen as large and portable as two opposing pages of a fake book thrown open on a music stand, I recommend investing in the physical realm.
By far, the most useful of the fake books I've owned has been Hal Leonard's Real Book (Sixth Edition). Every jazz jam I've attended has had at least one copy of this book (sometimes the pre-2004 illegal fifth edition) floating around. It's rare to see any other book, but there are other reasons to own one. In his article, Bob outlines several considerations like readability, self-study, and accuracy. He includes a spreadsheet of over 100 fake books with their various properties. (Given all my spreadsheeting habits, I can definitely nerd out on this.)
Once you've acquired a fake book or five, Bob and I highly recommend Seventh String's Fake Book Index. You can search any subset of fake books and play-alongs for different versions of the tune you want to study. Also check out the Rankings Indexes I created for the Real Book series.