When I try to get/install the Podcast app on the Touch I get a message that I need IOS 10.0 or higher. When I go to settings, it says I'm at IOS 6.x, and that I'm up to date. Any suggestions to either upgrade my IOS, or
iPod touch, iOS 6.1.6
iOS 10 is currently not actually released yet as its currently in Beta, and a 5th Gen iPod would not be able to update to it anyway. However it should be able to update to iOS 9.3.2.
If it is saying iOS 6.1.6 is up to date then you have a 4th Generation iPod touch, not a 5th generation one.
With that said however, you can try to get a compatible version of the App by first downloading the App to iTunes on a computer using the same Apple ID as the one being used on the iPod Touch.
When that is done you can follow this support document to a get a compatible version of the Podcasts App for iOS 6
Install an app on an earlier version of iOS or OS X - Apple Support
Podcast app on 5th gen iPod Touch
