How do I listen to music or radio while reading mail, etc?

If you are using the Music App, or say, Pandora, all you do start a song or album and while you are listening, hit your home button once to go to your home page. The music will cointinue playing and then you touch your mail icon and read mail. The music will continue playing while you're reading mail. Just remember to close the Music when you're finished. To do so, double tap yourhome button and you'll the most recently used apps (task bar) on the bottom. With your finger, hold it on any app until they all start to jiggle. You'll see ➖ on the upper left side. Touch the minus and the app will fully close.

If you're doing this through Safari, you won't be able to listen and read mail at the same time. You might want to see if Airkast offers an app in the App Store. There are radio apps like iHeart Radio, which if Airkast is on, you could open iHeart Radio, turn on Airkast, then do as I suggested and hit your home button, then open mail and you'd be good to go. But again, you won't be able to play a station through Safari and switch to another app.

It might be connected with Multitasking in iOS.

Lots of Apps are still not up-to-date with the multitasking feature in iOS (though launced more than a year ago).