Move all "All on My iPhone" contacts to "All Gmail" group
There is no easy way to move contacts between accounts. No company wants to make it too easy. The only option on iPhone would be to merge the On My iPhone contacts with an iCloud account (turning Contacts on and off if necessary under Settings > iCloud). 

From there you can log into iCloud.com > open the contacts web app > click the gear at the bottom left and select all > Click the gear at the bottom left and export vCard

Then you can import that vCard into gmail.com

Finally turn off the Contacts for iCloud and it will offer to delete.

Under Settings > Mail, Contacts and Calendars if there iOS a default account listed under contacts make sure it says gmail. 

Alternately - type lots.