Most eBooks you buy on Amazon, Kobo, etc, are protected by DRM (Digital Right Management) – meaning that you can not simply download your books and put them on any device you want.
First you'll need the excellent Calibre eBook management tool. Next you need to download Apprentice Alf's DeDRM plugin, the heart of this operation and the thing which will be doing the hard work for us. Remember to make sure to read the README and FAQ before you dive further into the stripping.
Uploading your own EPUB files to Google Play is about as easy as it gets. First you need to make sure your files are in the EPUB format. Convert them with Calibre if you need. Now you visit Google Play Books, and click "Upload files."
Worth noting is that Google Play is somewhat anal with the format of epub files. Download W3C's EPUBCheck and run the EPUB files through it in the case Google Play complains about being unable to process the files.
Just as with Google Play, you need to ensure your ebooks are in the EPUB format. So, if your books are in either MOBI/AZW3/PDF/etc, you need to first add your books to Calibre, then select to convert them into EPUB.
Connect your Kobo reader to your PC/MAC, then right-click the book(s) you want to transfer, and finally hit "Send to device."
If you so happen to own an Amazon Kindle, you have a couple of ways to upload your own eBooks to it.
First you may need to convert your files to one of the compatible formats. Amazon, in their attempt to lock you in, doesn't support the otherwise universally standard EPUB files, instead you need to have them in either MOBI, or AZW3 format. So, if your files are in some other format, fire up Calibre and click "Convert books," then select "AZW3" as the output format in the top right window.
Visit the Manage Your Kindle page and add your own email as an "Approved e-mail address." Next you just send yourself an email with the book as an attachment.
Just fire up Calibre, and plug in your device to your PC/Mac. Now you just select the books you want to upload, and right-click and select "Send to device."