Unless I'm not understanding the full scope of your problem, it should be as simple as highlighting your address book and clicking the "New Contact" button. (See attached image).

Editing an existing contact is done by choosing a contact and clicking the "Edit" button. (See attached image).

I'm aware of the process to add a name to my address book. I fill in the form and click Save and the name does not appear on the list. Also, I clicked the Edit button to change a different address, entered the new one, and the change does not appear on the list. The program is simply not working. Thanks for your help at this stage.

See https://support.mozilla.org/en-US/questions/1370420

I finally found the solution to 'how to add a new contact to a group'. Locate the address you want to add in your address book, left click and hold the click to drag that contact to the group in which it is to be added then drop the contact onto that group. It is then added to that group.