Thanks for reaching out, keenanmsc.
It sounds like you're having some difficulty with installing the Amazon Kindle app. We can help.
When you download an app for the first time, you will see the option to "Get" the app in the App Store. If you've downloaded the app previously, you'll see a cloud icon when downloading, but the download will work the same way. Amazon Kindle also may ask that some features be enabled to locate additional Amazon devices, but this should not impact your ability to read books on your iPhone.
Can you tell us more about what happens when you try to open and read a book in the Amazon Kindle app?
We'll go from there.