How to disable TouchWiz Easy Home mode and enable TouchWiz Home?

Here's how I got it to work on my S4.

There should be an entry on your apps (swipe to the right) called Easy Settings, which will have an Easy Mode button. If not, click More Apps, scroll to Settings, click "My Device", and then click Easy Mode.

From either of those, you can change it to Standard Mode.

If for some reason you can't access your settings or this doesn't work, one other option would be to download another home screen app from the Play Store, like Nova Launcher. When it installs, your phone will ask you which home screen you want to use when you press the home button. TouchWiz Easy Home and TouchWiz Home will be two separate options. You want TouchWiz Home.