Let Chappy, our token Android user, be your guide.
Okay, here’s the short answer for getting the .mobi to kindle (which is pretty easy, but still annoying because you can’t just search.)
Each device that the Kindle app is loaded on is given a mail-to-kindle email address. You find this address by looking at the settings.
Then simply attach the file to an email and send it to that address. The file magically shows up (if the device is connected to the internet) on the home page.
The only caveat is that the email you send it from has to be one that Amazon recognizes as yours. (This is so that people don’t randomly add stuff to your device) You can find out which addresses Amazon recognizes on your Amazon account under Manage Your Content and Devices.
(Another way is to move the file to the /kindle folder on the device, but this requires downloading a file manager.)