When I was going through a minor existential crisis a few years ago, I read the Bible from cover to cover. The copy I had was the rattiest, oldest paperback version I could find: I got it for two bucks at a library booksale.
The Bible app for iPhone and iPod touch, however, beats even that, because it's free. Bible is published by YouVersion.com, a social website for people reading the Bible that allows annotation and discussion of passages in the book.
Bible includes 13 English translations, three Spanish translations, and one translation each in French, German, and Simplified Chinese. It's searchable by term or location (chapter and verse), and includes a "daily read" tool that -- if you stick with it -- you can use to read the whole kit and kaboodle in a year.
There are more than 20 other Bible applications available in the App Store, costing as much as $30. The upshot is that some don't require an internet connection to view the text. (Thanks, commenters, for that one.) For example, Acro Bible NIV is $28.99, and includes only the New International Version. In this crowded category, there is also another app identically named "Bible," but it's $14.99.
There is also a free version of the Qur'an, called iQuran, available too. For the devout, there are web apps to help with prayer times as well, but no native iPhone app yet. As yet there isn't a full version of the Torah in the App Store, but there are both KosherMe (the mealtime blessings for $6.99) and Pocket Luach Lite (Jewish calendar conversion for $3.99).
Bible -- the YouVersion (uh) version -- is free, and available at the App Store.