It's not impossible, but the different companies and formats do make it difficult. The link mentioned above is a good start, though I'd suggest Calibre for converting ePub files into Mobis.
I read on both an iPad (Air 2) and a Kindle Paperwhite. Before the Air 2 I had a retina Mini.
My suggestion is to use the Kindle app for reading on the iPad, and get a Paperwhite with your own Amazon account for reading on a Kindle, then use wifi to sync between the two.