I want to be able to enter contact details of people from their email signatures into my address book automatically, without having to copy each phone number and name or other element one by one into each field.
You can also use a CRM (Customer Relationship Management) tool with your email. You can use many of them as plugins with your email client - gmail probably has the most options. Some like Capsule CRM have a nice button you click to sync up. You will, however, be committed to use the address book with the CRM.