Ok, you really do want the Kindle app first. That's probably where all your books are.
Amazon.com's Kindle reader is a huge hit. One of the things that makes it so popular, aside from access to a huge library of Kindle books on Amazon.com, is that Amazon.com offers an app for just about any devince you own, and it remembers where you left off from any Internet-connected device, so you can start reading on your iPod and finish on your Android. Now that isn't true of sideloaded books, but it is true of your Amazon purchases.
The thing to keep in mind as you build an Amazon.com library is that Amazon's books are meant to stay in Kindle readers. They use a proprietary format rather than keeping with industry-standard ePub format, and that locks you into staying with Amazon. It's by design.
2. The Nook App
The Nook Reader is Barns & Noble's baby. It's actually a pretty nice tablet, but it's still uses a modified version of Android that excludes you from Google Play. You're not locked into the Nook tablet to read Nook books. You can download the app and still access your library. Actually, my husband and I share libraries this way. He's in charge of paying for books on the Nook account, and I've got an Amazon account. It keeps the peace that way.
3. The Kobo AppThe Kobo reader was loosely affiliated with Borders, but fortunately not tightly enough to collapse when Borders did. Kobo offers a separate bookstore and books in ePUB format. However, it's at a disadvantage to the other more popular stores when it comes to content. It's actually superior to both of them when it comes to importing content. You can get separately purchased DRM-free books onto the Kobo reader with a lot less fussing than you can on the Nook or Kindle app. You also earn badges for reading and can track progress in books at a glance.
Formerly known as Google Books, Google Play Books is an app and a store. I recommend it last because it suffers some of the same content problems as the Kobo app and doesn't allow you to import books (at least not as of this writing. It's their most requested feature.) You can download your books and read them in some other reader. You'd have to convert them to mobi format in order to read them on a Kindle, but they're easy to import into your Kobo or Nook app.
The app itself is pleasant and easy to use. If Google would add the ability to import books you already own into your library, the same way you do with Google Music, it has the potential of suddenly being the best app of the bunch. The other killer feature Google could add would be the ability to sort and arrange books into "playlists" like you do with music. Now that my library is going all digital, I find that it's getting more cluttered than my physical bookshelves.