Default email app's notification is disabled on Samsung Galaxy S4
I had the same problem and this is what I did...
Start at your home screen and swipe down from the top to display your notifications. At the top right, tap the gear/wheel (which is settings). Then, tap accounts. Next, email. If you have more than one email on this app, you have to do the following for each email address--I have two on mine.

You will have a screen that has an accounts heading and common settings heading. Tap settings, then, on the next screen, tap the email you would like to adjust. Sync settings will come up. check the email notifications box and then select a ring tone that won't make you crazy every time you hear it! lol! And that should do it! 

p.s. you can also change your signature here as well, to personalize your outgoing messages by tapping signature and turning that function on..  :)