You get the Kindle app from iTunes on your PC or the App Store on your iPad, not from Amazon. Once you start the Kindle app, it will ask you to sign in and all you have to do is use the same username and password that you use for your Kindle. Once you do that, everything will be in your Archive and all you have to do is download them to your iPad in order to read them.
You can't use Calibre to convert Kindle books that you purchased to another format because they are protected by DRM. You have to use a Kindle or the Kindle app unless you unwrap the DRM.... which is something we do not discuss on this forum!