AltaMail has 2 ways of creating your own contact groups. When viewing the contact screen, tap on the group button in the top of the screen. Tap on the wrench button in the top of the group screen to show the action menu button where you can choose what type of new group to create.
The "New Device Contact Group" will create a group that is available to your Apple Contacts app, as well as other apps that use contacts.
The "New AltaMail Contact Group" will create a group that is only visible inside AltaMail.

To delete a contact group, tap on the group button which selects groups and then tap on the wrench button in the top right. Then you can select groups with the checkbox and use the action menu to rename or delete groups.

You can create a new contact in the "All Contacts" group by tapping on the "+" button. You can then enter the name and a new contact will be created. Tap on the new contact to add other details, such as phone and email.
For other contact groups, use the "+" button in the top of the screen for adding contacts from other groups. These contacts will still exist in other groups too so changes to these contacts will change the details in all group locations.

To permanently delete a contact from the "All Contacts" group, tap on the checkbox beside the contact and choose "Delete" in the action menu (circle with 3 dots button).
To remove a contact from other groups, tap on the checkbox beside the contact and choose "Remove contact" in the action menu.