I use Calibre
to administer my books. So my starting point is always adding new books to my Calibre library.
However when I want to send a book to my Kindle or to the Kindle app on my iPhone I use the Amazon app, Send to Kindle
From Calibre, I invoke the Save to Disk function (MOBI format), then I right-click the copy I have just saved and select Send to Kindle. As I have both Kindle and the Kindle app on an iPhone, I have set up Send to Kindle to send to both devices at once. This process works well for me.