How can I set up email on my GS4 to handle both email and gmail accounts?
This official Google support page tells you how to access Gmail from a normal email client using IMAP: you have to turn it on through the Gmail web interface, then add the account to your email client like any other IMAP server.

The standard email client on Samsung devices supports having multiple accounts. Press the + button in the Email app's settings to add an additional account. Most third-party email clients for Android (such as K9-Mail) support multiple accounts too, but the set-up will be slightly different for each.