With Google and Yahoo updating their mail apps and all the excitement surrounding the recently released Mailbox app on iOS I figured I'd explain how to setup your iPhone to use other mail clients.
While Apple hasn't officially allowed iOS users to fully replace their standard Mail app (or any other of the default apps) there are ways to set it up your iPhone where you rarely have to use it.
The main thing you want to do is turn off notifications of the default Mail app and turn on the notifications of your new Mail client. You want to do this so you won't get a notification from both apps whenever an email arrives.
1. Go to your Settings App and select Notifications
2. Once in Notifications, Find the default Mail app (very bottom of image)
3. Select Mail app and turn OFF Notification Center, set Alert Style to NONE, Badge App Icon to OFF and View In Lock Screen to OFF as well. Select New Mail Sound and set Alert Tone to NONE and Vibration to NONE.
4. Once all of your Notifications and alerts are turned off for the old Mail app go back to the Notification area and select your new Mail app (Gmail, Mailbox, etc.)
5. Turn on Notification Center and adjust the Notifications and alerts to your liking and you're done!
Now whenever you get a new email the Notification should be from the new app only. You can't delete the default Apple Mail app so you're just going to have to hide it in a folder with all the other apps you don't use but can't delete. The only time you'll have to see or use the Apple Mail app is when you click on an email link in a web browser, contacts, etc.