I have been successful in managing contact groups with OS X's Contacts.app and iCloud.com, but haven't found a way to do this as of iOS 9.
Is there a way to manage contact groups on iOS (iPhone, iPad, or iPod Touch) ?
No, there is no way to create or edit contact groups on iOS. I share your frustration! The best workaround is to use iCloud.com (or your primary Mac, if you have it with you).
Also, there is a way to add contacts to a specific group on iOS, though it's not obvious:
Now when you create and save the contact, it'll save it in that specific group. You can then go back to the Groups list and reenable all your groups.
The answer is simple and yet dissatisfactioning: NO (with standard apps).
EDIT: There are some apps on the AppStore which enable this feature. One of them is "Groups Free". At least it adds groups to my iCloud account.