With Android, at least Moon+ Reader (which is my personal choice) and FBReader can connect to an OPDS feed from Calibre. However you probably want to read
this sticky for a list of lots of options.
Alternatively, using the
Calibre Companion allows you to connect to your Calibre library and transfer books. As long as your reading app allows you to scan a local folder you'll be all sorted.
Finally, for iOS see
this sticky.